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Solutions

Solutions

1.

 

a. First determine the eigenvalues:

                        det = 0

                        (-1 - l)(2 - l) - 22 = 0

                        -2 + l - 2l + l2 - 4 = 0

                        l2 - l - 6 = 0

                        (l - 3)(l + 2) = 0

                        l = 3    or    l = -2.

Next, determine the eigenvectors.  First, the eigenvector associated with eigenvalue -2:

                        = -2

                        -C11 + 2C21 = -2C11

C11 = -2C21  (Note: The second row offers no new information, e.g. 2C11 + 2C21 = -2C21)

                        C112 + C212 = 1  (from normalization)

                        (-2C21)2 + C212 = 1

                        4C212 + C212 = 1

                        5C212 = 1

                        C212 = 0.2

                        C21 = , and therefore C11 = -2.

For the eigenvector associated with eigenvalue 3:

                        = 3

                        -C12 + 2C22 = 3C12

                                -4C12 = -2C22

                                C12 = 0.5C22  (again the second row offers no new information)

                        C122 + C222 = 1  (from normalization)

                        (0.5C22)2 + C222 = 1

                        0.25C222 + C222 = 1

                        1.25C222 = 1

                        C222 = 0.8

                        C22 = = 2, and therefore C12 = .

Therefore the eigenvector matrix becomes:

                       

            b. First determine the eigenvalues:

                        det = 0

                        det det = 0

From 1a, the solutions then become -2, -2, and 3.  Next, determine the eigenvectors.  First the eigenvector associated with eigenvalue 3 (the third root):

                        = 3

                        -2 C13 = 3C13  (row one)

                        C13 = 0

                        -C23 + 2C33 = 3C23  (row two)

                        2C33 = 4C23

                        C33 = 2C23  (again the third row offers no new information)

                        C132 + C232 + C332 = 1  (from normalization)

                        0 + C232 + (2C23)2 = 1

                        5C232 = 1

                        C23 = , and therefore C33 = 2.

Next, find the pair of eigenvectors associated with the degenerate eigenvalue of -2.  First, root one eigenvector one:

                        -2C11 = -2C11  (no new information from row one)

                        -C21 + 2C31 = -2C21  (row two)

                        C21 = -2C31  (again the third row offers no new information)

                        C112 + C212 + C312 = 1  (from normalization)

                        C112 + (-2C31)2 + C312 = 1

                        C112 + 5C312 = 1

C11 =

Second, root two eigenvector two:

                        -2C12 = -2C12  (no new information from row one)

                        -C22 + 2C32 = -2C22  (row two)

                        C22 = -2C32  (again the third row offers no new information)

                        C122 + C222 + C322 = 1  (from normalization)

                        C122 + (-2C32)2 + C322 = 1

                        C122 + 5C322 = 1

C12 = (1- 5C322)1/2 (Note: again, two equations in three unknowns)

                        C11C12 + C21C22 + C31C32 = 0  (from orthogonalization)

Now there are five equations with six unknowns. 

                        Arbitrarily choose C11 = 0

(whenever there are degenerate eigenvalues, there are not unique eigenvectors because the degenerate eigenvectors span a 2- or more- dimensional space, not two unique directions. One always is then forced to choose one of the coefficients and then determine all the rest; different choices lead to different final eigenvectors but to identical spaces spanned by these eigenvectors).

                        C11 = 0 =

                        5C312 = 1

                        C31 =

                        C21 = -2

                        C11C12 + C21C22 + C31C32 = 0  (from orthogonalization)

                        0 + -2+ C32 = 0

                        5C32 = 0

                        C32 = 0, C22 = 0, and C12 = 1

Therefore the eigenvector matrix becomes:

                       

 

 

 

2.

a.         K.E. = = = =

                        K.E. =

                        K.E. =

                        K.E. =

            b.         p = mv = ipx + jpy + kpz 

                        p =

                        where i, j, and k are unit vectors along the x, y, and z axes.

            c.         Ly = zpx - xpz

                        Ly = z - x

 

 

3.

First derive the general formulas for , , in terms of r,q, and f, and , , and in terms of x,y, and z.  The general relationships are as follows:

                        x = r Sinq Cosf                       r2 = x2 + y2 + z2

                        y = r Sinq Sinf                        sinq =

                        z = r Cosq                                cosq =

                                                                        tanf =

 

First , , and from the chain rule:

            = y,z + y,z + y,z ,

            = x,z + x,z + x,z ,

            = x,y + x,y + x,y .

Evaluation of the many "coefficients" gives the following:

            y,z = Sinq Cosf , y,z = , y,z = - ,

            x,z = Sinq Sinf , x,z = , x,z = ,

            x,y = Cosq , x,y = - , and  x,y = 0 .

Upon substitution of these "coefficients":

            = Sinq Cosf + - ,

            = Sinq Sinf + + , and

            = Cosq - + 0 .

Next , , and from the chain rule:

            = q,f + q,f + q,f ,

            = r,f + r,f + r,f , and

            = r,q + r,q + r,q .

Again evaluation of the the many "coefficients" results in:

            q,f = , q,f = ,

            q,f = , r,f = , r,f = ,

            r,f = - , r,q = -y , r,q = x , and r,q = 0

Upon substitution of these "coefficients":

            = +

                         +

            = + -

            = -y + x + 0 .

Note, these many "coefficients" are the elements which make up the Jacobian matrix used whenever one wishes to transform a function from one coordinate representation to another.  One very familiar result should be in transforming the volume element dxdydz to r2Sinqdrdqdf.  For example:

            =

           

            a.         Lx =

                        Lx =

                         -

                        Lx = -

            b.         Lz = = - i

                        Lz =

 

 

4.

 

                        B                                 dB/dx                          d2B/dx2          

            i.          4x4 - 12x2 + 3             16x3 - 24x                   48x2 - 24

            ii.         5x4                              20x3                            60x2

            iii.        e3x + e-3x                    3(e3x - e-3x)     9(e3x + e-3x)

            iv.        x2 - 4x + 2                   2x - 4               2

            v.         4x3 - 3x                       12x2 - 3                       24x

B(v.) is an eigenfunction of A(i.):

            (1-x2) - x B(v.) =

            (1-x2) (24x) - x (12x2 - 3)

            24x - 24x3 - 12x3 + 3x

            -36x3 + 27x

            -9(4x3 -3x)  (eigenvalue is -9)

B(iii.) is an eigenfunction of A(ii.):

            B(iii.) =

            9(e3x + e-3x)  (eigenvalue is 9)

B(ii.) is an eigenfunction of A(iii.):

            x B(ii.) =

            x (20x3)

            20x4

            4(5x4)  (eigenvalue is 4)

B(i.) is an eigenfunction of A(vi.):

            - 2x B(i) =

            (48x2 - 24) - 2x (16x3 - 24x)

            48x2 - 24 - 32x4 + 48x2

            -32x4 + 96x2 - 24

            -8(4x4 - 12x2 + 3)  (eigenvalue is -8)

B(iv.) is an eigenfunction of A(v.):

            x + (1-x) B(iv.) =

            x (2) + (1-x) (2x - 4)

            2x + 2x - 4 - 2x2 + 4x

            -2x2 + 8x - 4

            -2(x2 - 4x +2)  (eigenvalue is -2)

 

 

 

5.

 

 

 

                       

 

                       

 

                   

 

 

 

 

 

 

 

 

6.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

7.

 

 

 

 

 

 

 

8.

 

i. In ammonia, the only "core" orbital is the N 1s and this becomes an a1 orbital in C3v symmetry.  The N 2s orbitals and 3 H 1s orbitals become 2 a1 and an e set of orbitals.  The remaining N 2p orbitals also become 1 a1 and a set of e orbitals.  The total valence orbitals in C3v symmetry are 3a1 and 2e orbitals.

ii. In water, the only core orbital is the O 1s and this becomes an a1 orbital in C2v symmetry.  Placing the molecule in the yz plane allows us to further analyze the remaining valence orbitals as: O 2pz = a1, O 2py as b2, and O 2px as b1.  The (H 1s + H 1s) combination is an a1 whereas the (H 1s - H 1s) combination is a b2.

iii. Placing the oxygens of H2O2 in the yz plane (z bisecting the oxygens) and the (cis) hydrogens distorted slightly in +x and -x directions allows us to analyze the orbitals as follows.  The core O 1s + O 1s combination is an a orbital whereas the O 1s - O 1s combination is a b orbital.  The valence orbitals are: O 2s + O 2s = a, O 2s - O 2s = b, O 2px + O 2px = b, O 2px - O 2px = a, O 2py + O 2py = a, O 2py - O 2py = b, O 2pz + O 2pz = b, O 2pz - O 2pz = a, H 1s + H 1s = a, and finally the H 1s - H 1s = b.

 

iv. For the next two problems we will use the convention of choosing the z axis as principal axis for the D¥h, D2h, and C2v point groups and the xy plane as the horizontal reflection plane in Cs symmetry.

                                                D¥h                 D2h                  C2v                  Cs

                        N 1s                 sg                           ag                     a1                     a'

                        N 2s                 sg                           ag                     a1                     a'

                        N 2px                   pxu                        b3u                   b1                    a'

                        N 2py                   pyu                  b2u                   b2                    a'

                        N 2pz                   su                    b1u                   a1                     a''

 

 

9.

 

            a.         Yn(x) = Sin

                        Pn(x)dx = dx

The probability that the particle lies in the interval 0 £ x £ is given by:

                        Pn = =

This integral can be integrated to give :

                        Pn =

                        Pn =

                        Pn = - \f(1,4q,2

                                    =

                                    = - Sin

            b. If n is even, Sin= 0 and Pn = .

                        If n is odd and n = 1,5,9,13, ... Sin= 1

                        and Pn = -

                        If n is odd and n = 3,7,11,15, ... Sin= -1

                        and Pn = +

                        The higher Pn is when n = 3.  Then Pn = +

                        Pn = + = 0.303

            c. Y(t) = e= aYne+ bYme

                        HY = aYnEne+ bYmEme

                        = |a|2En + |b|2Em + a*be

                                                + b*ae

                        Since and are zero,

                        = |a|2En + |b|2Em  (note the time independence)

            d. The fraction of systems observed in Yn is |a|2.  The possible energies measured are En and Em.  The probabilities of measuring each of these energies is |a|2 and |b|2.

            e. Once the system is observed in Yn, it stays in Yn.

            f. P(En) = 2 = |cn|2

            cn = x(L-x)dx

                            = dx

                            =

These integrals can be evaluated to give:

                        cn = 60,L6L\b(\f(L2,n2p2

                        - 60,L6\b(\f(2xL2,n2p2

                        cn = {

                                    - )

                                    - (

                                    - Cos(np)

                                    + Cos(0))}

                        cn = L-3{- Cos(np) + Cos(np)

                                    + }

                        cn =

                        cn =

                        cn = )

                        |cn|2 = )

                        If n is even then cn = 0

                        If n is odd then cn = =

The probability of making a measurement of the energy and obtaining one of the eigenvalues, given by:

                        En = is:

                        P(En) = 0 if n is even

                        P(En) = if n is odd

            g. =

                                     =

                                     =

                                     =

                                     = 30\o(h,-L\f(x2,2

                                     =

                                     =

                                     = =

 

10.

 

= Ci*eeCj

            Since = Ejdij

            = Cj*CjEje

            =

For other properties:

            = Ci*eeCj

but, does not necessarily = ajdij because the Yj  are not eigenfunctions of A unless [A,H] = 0.

            = Ci*Cje

Therefore, in general, other properties are time dependent.

 

 

11.

 

a. The lowest energy level for a particle in a 3-dimensional box is when n1 = 1, n2 = 1, and n3 = 1.  The total energy (with L1 = L2 = L3) will be:

                        Etotal = =

Note that n = 0 is not possible.  The next lowest energy level is when one of the three quantum numbers equals 2 and the other two equal 1:

                                    n1 = 1, n2 = 1, n3 = 2

                                    n1 = 1, n2 = 2, n3 = 1

                                    n1 = 2, n2 = 1, n3 = 1.

Each of these three states have the same energy:

                        Etotal = =

Note that these three states are only degenerate if L1 = L2 = L3.

 

            b.         ¾  ¾¾  ¾¾                                      ¾¾  ¾¾

                                                                                                     ¾

                                                                                          

                        L1 = L2 = L3                                                           L3 ¹ L1 = L2

 

For L1 = L2 = L3, V = L1L2L3 = L13,

Etotal(L1) = 2e1 + e2

            = +

            = + =

For L3 ¹ L1 = L2, V = L1L2L3 = L12L3, L3 = V/L12

Etotal(L1) = 2e1 + e2

            = +

            = +

            =

            = =

In comparing the total energy at constant volume of the undistorted box (L1 = L2 = L3) versus the distorted box (L3 ¹ L1 = L2) it can be seen that:

            £ as long as L3 ³ L1.

            c. In order to minimize the total energy expression, take the derivative of the energy with respect to L1 and set it equal to zero.                       = 0

                        = 0

But since V = L1L2L3 = L12L3, then L3 = V/L12.  This substitution gives:

                        = 0

                        = 0

                        = 0

                        =

                        24L16 = 12V2

                        L16 = V2 = = L14L32

                        L12 = L32

                        L3 = L1

            d. Calculate energy upon distortion:

cube:    V = L13, L1 = L2 = L3 = (V)

distorted:           V = L12L3 = L12L1 = L13

                        L3 = ¹ L1 = L2 =

DE = Etotal(L1 = L2 = L3) - Etotal(L3 ¹ L1 = L2)

            = -

            =

            =

Since V = 8Å3, V2/3 = 4Å2 = 4 x 10-16 cm2 , and = 6.01 x 10-27 erg cm2:

            DE = 6.01 x 10-27 erg cm2

            DE = 6.01 x 10-27 erg cm2

            DE = 0.99 x 10-11 erg

            DE = 0.99 x 10-11 erg

            DE = 6.19 eV

 

 

12.

 

a.         H =   (Cartesian coordinates)

Finding andfrom the chain rule gives:

            = y + y ,  = x + x ,

Evaluation of the "coefficients" gives the following:

            y = Cosf , y = - ,

            x = Sinf , and  x = ,

Upon substitution of these "coefficients":

            = Cosf - = - ; at fixed r.

            = Sinf + = ; at fixed r.

            =

                  = + ; at fixed r.

            =

                  = - ; at fixed r.

            + = + + -

                              = ; at fixed r.

            So,       H = (cylindrical coordinates, fixed r)

                           =

The Schrödinger equation for a particle on a ring then becomes:

            HY = EY

            = EF

            = F

The general solution to this equation is the now familiar expression:

            F(f) = C1e-imf + C2eimf , where m =

Application of the cyclic boundary condition, F(f) = F(f+2p), results in the quantization of the energy expression: E = where m = 0, ±1, ±2, ±3, ...  It can be seen that the ±m values correspond to angular momentum of the same magnitude but opposite directions.  Normalization of the wavefunction (over the region 0 to 2p) corresponding to + or - m will result in a value of for the normalization constant.

                        \ F(f) = eimf

 

                        ¾¾  ¾¾            

                        ¾¾  ¾¾            

                        ¾¾  ¾¾            

                          

                                

            b. = 6.06 x 10-28 erg cm2 

            =

                        = 3.09 x 10-12 erg

            DE = (22 - 12) 3.09 x 10-12 erg = 9.27 x 10-12 erg

but DE = hn = hc/l     So l = hc/DE

            l =

               = 2.14 x 10-5 cm = 2.14 x 103 Å

Sources of error in this calculation include:

            i. The attractive force of the carbon nuclei is not included in the Hamiltonian.

            ii. The repulsive force of the other p-electrons is not included in the Hamiltonian.

            iii. Benzene is not a ring.

            iv. Electrons move in three dimensions not one.

           

 

13.

 

Y(f,0) = Cos2f.

This wavefunction needs to be expanded in terms of the eigenfunctions of the angular momentum operator, .  This is most easily accomplished by an exponential expansion of the Cos function.

            Y(f,0) =

                         =

The wavefunction is now written in terms of the eigenfunctions of the angular momentum operator, , but they need to include their normalization constant, .

            Y(f,0) =

                         =

Once the wavefunction is written in this form (in terms of the normalized eigenfunctions of the angular momentum operator having mas eigenvalues) the probabilities for observing angular momentums of 0, 2, and -2can be easily identified as the squares of the coefficients of the corresponding eigenfunctions.

                        P2= =

                        P-2= =

                        P0= =

 

14.

 

a. mv2 = 100 eV

                        v2 =

                        v = 0.593 x 109 cm/sec

The length of the N2 molecule is 2Å = 2 x 10-8 cm.

                        v =

                        t = = = 3.37 x 10-17 sec

            b. The normalized ground state harmonic oscillator can be written as:

                        Y0 = 1/4e-ax2/2,    where a = and x = r - re

Calculating constants;

            aN2 =

                        = 0.48966 x 1019 cm-2 = 489.66 Å-2

For N2:            Y0(r) = 3.53333Åe-(244.83Å-2)(r-1.09769Å)2

            aN2+ =

                        = 0.45823 x 1019 cm-2 = 458.23 Å-2

For N2+:          Y0(r) = 3.47522Åe-(229.113Å-2)(r-1.11642Å)2

            c. P(v=0) =

Let P(v=0) = I2               where I = integral:

            I= .

                                    (3.53333Åe-(244.830Å-2)(r-1.09769Å)2)dr

Let       C1 = 3.47522Å,      C2 = 3.53333Å,

            A1 = 229.113Å-2,          A2 = 244.830Å-2,      

            r1 = 1.11642Å,                        r2 = 1.09769Å,                        

            I = C1C2dr .

Focusing on the exponential:

-A1(r-r1)2-A2(r-r2)2 = -A1(r2 - 2r1r + r12) - A2(r2 - 2r2r + r22)

                                         = -(A1 + A2)r2 + (2A1r1 + 2A2r2)r - A1r12 - A2r22

Let       A = A1 + A2,

            B = 2A1r1 + 2A2r2,

            C = C1C2, and

            D = A1r12 + A2r22 .

            I = Cdr

               = Cdr

where -A(r-r0)2 + D' = -Ar2 + Br - D

                        -A(r2 - 2rr0 + r02) + D' = -Ar2 + Br - D

such that,         2Ar0 = B

                        -Ar02 + D' = -D

and,                 r0 =

                        D' = Ar02 - D = A- D = - D .

                        I = Cdr

                           = CeD'dy

                           = CeD'

Now back substituting all of these constants:

I = C1C2exp

I = (3.47522)(3.53333)

                        . exp

                         . exp

                        I = 0.959

                        P(v=0) = I2 = 0.92, so there is a 92% probability.

 

15.

 

a.         En =

                        DE = En+1 - En

                             = =

                             =

                             = 4.27 x 10-13 erg

                        DE =

                        l = =

                           = 4.66 x 10-4 cm

                        = 2150 cm-1

            b.         Y0 = 1/4e-ax2/2

                        =

                                     =

                                     =

                                     =

                                     = 1/2e-ax2 ½= 0

                        =

                                     =

                                     =

                                     = 21/2

                                     = 21/21/2

                                     =

                        Dx = (<x2> - <x>2)1/2.=

                             =

                             =

                             = 3.38 x 10-10 cm = 0.0338Å

            c.         Dx =

The smaller k and m become, the larger the uncertainty in the internuclear distance becomes.  Helium has a small m and small attractive force between atoms.  This results in a very large Dx.  This implies that it is extremely difficult for He atoms to "vibrate" with small displacement as a solid, even as absolute zero is approached.

 

 

16.

a.         W =

                        W =

                        e=

                                           = +

                                           = +

 

Making this substitution results in the following three integrals:

            W = +

                        +

                       

               = + +

                        a

               = 2 + 2 +

                        a

               = + +

            W = + a

            b. Optimize b by evaluating = 0

                        =

                                    = - b

So,  b= or, b= = ,

and, b = .  Substituting this value of b into the expression for W gives:

            W = + a

                = + a

                = 2pam+ 2pam

                = am= am

                = 0.812889106am-1/3 which is in error by only 0.5284% !!!!!

 

 

17.

 

a.         H = -+ kx2

                        f = a          for -a < x < a

                        f = 0                            for |x| ³ a

 =

 = a-5

 = a-5

                                     + a-5

 = a-5

                                     + a-5

 = a-5dx + a-5

 = a-5\o(\s\up10(a-a

                         + a-5a4k,3\o(\s\up10(a

 = a-5+ a-5

 = a-5

 = a-5

 = a-5

 = a-5= +

            b. Substituting a = binto the above expression for E we obtain:

                        E = +

                           = km

 

            c.         E = +

                        = -+ = -+ = 0

                        = and 352 = 2mka4 

                         So, a4 = , or a =

Therefore fbest = ,

and Ebest = + = km.

            d. =

                                     = = = 0.1952 = 19.52%

 

18.

 

a.         H0 y= y= Yl,m(q,f)

                                       = 2 l(l+1) Yl,m(q,f)

                        E= l(l+1)

            b.         V = -eez = -eer0Cosq

                        E= =

                               = -eer0

Using the given identity this becomes:

            E= -eer0+

                                                -eer0

The spherical harmonics are orthonormal, thus = = 0, and E= 0.

            E=

            = -eer0

Using the given identity this becomes:

            = -eer0+

                                                -eer0

            = -

This indicates that the only term contributing to the sum in the expression for Eis when l=1, and m=), otherwise vanishes (from orthonormality).  In quantum chemistry when using orthonormal functions it is typical to write the term as a delta function, for example dlm,10 , which only has values of 1 or 0; dij = 1 when i = j and 0 when i ¹ j.  This delta function when inserted into the sum then eliminates the sum by "picking out" the non-zero component.  For example,

            = -dlm,10 ,         so

            E= = 

            E= 0(0+1) = 0 and E= 1(1+1) =

Inserting these energy expressions above yields:

            E= -= -

            c.         E= E+ E+ E+ ...

                                = 0 + 0 -

                                = -

                        a = -=

                            =

            d.         a =

                        a = r04 12598x106cm-1 = r04 1.2598Å-1

                        aH = 0.0987 Å3

                        aCs = 57.57 Å3

 

 

19.

 

 

 

The above diagram indicates how the SALC-AOs are formed from the 1s,2s, and 2p N atomic orbitals.  It can be seen that there are 3sg, 3su, 1pux, 1puy, 1pgx, and 1pgy SALC-AOs.  The Hamiltonian matrices (Fock matrices) are given.  Each of these can be diagonalized to give the following MO energies:

            3sg; -15.52, -1.45, and -0.54 (hartrees)

            3su; -15.52, -0.72, and 1.13

            1pux; -0.58

            1puy; -0.58

            1pgx; 0.28

            1pgy; 0.28

It can be seen that the 3sg orbitals are bonding, the 3su orbitals are antibonding, the 1pux and 1puy orbitals are bonding, and the 1pgx and 1pgy orbitals are antibonding.

20.

Using these approximate energies we can draw the following MO diagram:

                       

This MO diagram is not an orbital correlation diagram but can be used to help generate one.  The energy levels on each side (C and H2) can be "superimposed" to generate the reactant side of the orbital correlation diagram and the center CH2 levels can be used to form the product side.  Ignoring the core levels this generates the following orbital correlation diagram.

           

21.

                       

            a. The two F p orbitals (top and bottom) generate the following reducible representation:

                        D3h      E  2C3  3C2   sh  2S3  3sv

                        Gp        2    2     0     0     0     2 

This reducible representation reduces to 1A1' and 1A2''  irreducible representations.

Projectors may be used to find the symmetry-adapted AOs for these irreducible representations.

                        fa1' =

                        fa2'' =

            b. The three trigonal F p orbitals generate the following reducible representation:

                        D3h      E  2C3  3C2   sh  2S3  3sv

                        Gp        3    0     1     3     0     1 

This reducible representation reduces to 1A1' and 1E'  irreducible representations.

Projectors may be used to find the symmetry-adapted -AOs for these irreducible representations (but they are exactly analogous to the previous few problems):

                        fa1' =

                        fe' = (1/6)-1/2 (2 f3 – f4 –f5)

                        fe' = .

            c. The 3 P sp2 orbitals generate the following reducible representation:

                        D3h      E  2C3  3C2   sh  2S3  3sv

                        Gsp2      3     0        1     3    0     1 

This reducible representation reduces to 1A1' and 1E'  irreducible representations.  Again, projectors may be used to find the symmetry-adapted -AOs for these irreducible representations:

                        fa1' =

                        fe' =

                        fe' = .

The leftover P pz orbital generate the following irreducible representation:

                        D3h      E  2C3  3C2   sh  2S3  3sv

                        Gpz      1    1    -1    -1    -1     1 

This irreducible representation is  A2''

                        fa2'' = f9.

 

Drawing an energy level diagram using these SALC-AOs would result in the following:

                                                           

 

 

22.

 

a. For non-degenerate point groups, one can simply multiply the representations (since only one representation will be obtained):

                         a1 Ä b1 = b1

Constructing a "box" in this case is unnecessary since it would only contain a single row.  Two unpaired electrons will result in a singlet (S=0, MS=0), and three triplets (S=1, MS=1; S=1, MS=0; S=1, MS=-1).  The states will be: 3B1(MS=1), 3B1(MS=0), 3B1(MS=-1), and 1B1(MS=0).

b. Remember that when coupling non-equivalent linear molecule angular momenta, one simple adds the individual Lz values and vector couples the electron spin.  So, in this case (1pu12pu1), we have ML values of 1+1, 1-1, -1+1, and -1-1 (2, 0, 0, and -2).  The term symbol D is used to denote the spatially doubly degenerate level (ML=±2) and there are two distinct spatially non-degenerate levels denoted by the term symbol S (ML=0)  Again, two unpaired electrons will result in a singlet (S=0, MS=0), and three triplets (S=1, MS=1;S=1, MS=0;S=1, MS=-1).  The states generated are then:

                        1D (ML=2); one state (MS=0),

                        1D (ML=-2); one state (MS=0),

                        3D (ML=2); three states (MS=1,0, and -1),

                        3D (ML=-2); three states (MS=1,0, and -1),

                        1S (ML=0); one state (MS=0),

                        1S (ML=0); one state (MS=0),

                        3S (ML=0); three states (MS=1,0, and -1), and

                        3S (ML=0); three states (MS=1,0, and -1).

            c. Constructing the "box" for two equivalent p electrons one obtains:

 

                 ML

MS

2

1

0

1

 

 

|p1ap-1a|

0

|p1ap1b|

 

|p1ap-1b|,

|p-1ap1b|

 

From this "box" one obtains six states:

                        1D (ML=2); one state (MS=0),

                        1D (ML=-2); one state (MS=0),

                        1S (ML=0); one state (MS=0),

                        3S (ML=0); three states (MS=1,0, and -1).

 

            d. It is not necessary to construct a "box" when coupling non-equivalent angular momenta since vector coupling results in a range from the sum of the two individual angular momenta to the absolute value of their difference.  In this case, 3d14d1, L=4, 3, 2, 1, 0, and S=1,0.  The term symbols are: 3G, 1G, 3F, 1F, 3D, 1D, 3P, 1P, 3S, and 1S.  The L and S angular momenta can be vector coupled to produce further splitting into levels:

                        J = L + S ... |L - S|.

Denoting J as a term symbol subscript one can identify all the levels and subsequent (2J + 1) states:

                        3G5 (11 states),

                        3G4 (9 states),

                        3G3 (7 states),

                        1G4 (9 states),

                        3F4 (9 states),

                        3F3 (7 states),

                        3F2 (5 states),

                        1F3 (7 states),

                        3D3 (7 states),

                        3D2 (5 states),

                        3D1 (3 states),

                        1D2 (5 states),

                        3P2 (5 states),

                        3P1 (3 states),

                        3P0 (1 state),

                        1P1 (3 states),

                        3S1 (3 states), and

                        1S0 (1 state).

 

            e. Construction of a "box" for the two equivalent d electrons generates (note the "box" has been turned side ways for convenience):

 

                         MS

ML

1

0

4

 

|d2ad2b|

3

|d2ad1a|

|d2ad1b|,         |d2bd1a|

2

|d2ad0a|

|d2ad0b|,         |d2bd0a|, |d1ad1b|

1

|d1ad0a|,         |d2ad-1a|

|d1ad0b|,         |d1bd0a|, |d2ad-1b|,        |d2bd-1a|

0

|d2ad-2a|,        |d1ad-1a|

|d2ad-2b|,        |d2bd-2a|, |d1ad-1b|,        |d1bd-1a|, |d0ad0b|

 

The term symbols are: 1G, 3F, 1D, 3P, and 1S.  The L and S angular momenta can be vector coupled to produce further splitting into levels:

                        1G4 (9 states),

                        3F4 (9 states),

                        3F3 (7 states),

                        3F2 (5 states),

                        1D2 (5 states),

                        3P2 (5 states),

                        3P1 (3 states),

                        3P0 (1 state), and

                        1S0 (1 state).

 

23.

 

a. Once the spatial symmetry has been determined by multiplication of the irreducible representations, the spin coupling gives the result:

                       

            b. There are three states here :

                        1.) |3a1a1b1a|,

                        2.) , and

                        3.) |3a1b1b1b|

            c. |3a1a3a1b|

 

 

24.

 

a. All the Slater determinants have in common the |1sa1sb2sa2sb| "core" and hence this component will not be written out explicitly for each case.

3P(ML=1,MS=1)         = |p1ap0a|

                                    = |a(pz)a|

                                    =

3P(ML=0,MS=1)         = |p1ap-1a|

                                    = |aa|

                                    =

                                    =

                                    =

                                    = -i|pxapya|

3P(ML=-1,MS=1)        = |p-1ap0a|

                                    = |a(pz)a|

                                    =

As you can see, the symmetries of each of these states cannot be labeled with a single irreducible representation of the C2v point group.  For example, |pxapza| is xz (B1) and |pyapza| is yz (B2) and hence the 3P(ML=1,MS=1) state is a combination of B1 and B2 symmetries.  But, the three 3P(ML,MS=1) functions are degenerate for the C atom and any combination of these three functions would also be degenerate.  Therefore, we can choose new combinations that can be labeled with "pure" C2v point group labels.

3P(xz,MS=1) = |pxapza|

                         = = 3B1

3P(yx,MS=1) = |pyapxa|

                          = = 3A2

3P(yz,MS=1) = |pyapza|

                          = = 3B2

Now, we can do likewise for the five degenerate 1D states:

1D(ML=2,MS=0)         = |p1ap1b|

                                    = |ab|

                                    =

1D(ML=-2,MS=0)       = |p-1ap-1b|

                                    = |ab|

                                    =

1D(ML=1,MS=0)         =

                                    =

                                    =

1D(ML=-1,MS=0)       =

                                    =

                                    =

1D(ML=0,MS=0)         =

                                    =

                                     + |ab|)

                                    =

                                     +

                                     + )

                                    = )

Analogous to the three 3P states, we can also choose combinations of the five degenerate 1D states which can be labeled with "pure" C2v point group labels:

1D(xx-yy,MS=0) = |pxapxb| - |pyapyb|

                         = = 1A1

1D(yx,MS=0) = |pxapyb| + |pyapxb|

                         = = 1A2

1D(zx,MS=0) = |pzapxb| - |pzbpxa|

                          = = 1B1

1D(zy,MS=0) = |pzapyb| - |pzbpya|

                          = = 1B2

1D(2zz+xx+yy,MS=0) = )

                          = 1D(ML=0,MS=0) = 1A1

The only state left is the 1S:

1S(ML=0,MS=0)         =

                                    =

                                     - |ab|)

                                    =

                                     -

                                     - )

                                    = )

Each of the components of this state are A1 and hence this state has

A1 symmetry.

 

            b. Forming symmetry-adapted AOs from the C and H atomic orbitals would generate the following:

 

 

The bonding, nonbonding, and antibonding orbitals of CH2 can be illustrated in the following manner:

 

           

 

            c.

                       

            d. - e. It is necessary to determine how the wavefunctions found in part a. correlate with states of the CH2 molecule:

                                3P(xz,MS=1); 3B1 = sg2s2pxpz ¾¾® s2n2pps*

                                3P(yx,MS=1); 3A2 = sg2s2pxpy ¾¾® s2n2pps

                                3P(yz,MS=1); 3B2 = sg2s2pypz ¾¾® s2n2ss*

                                1D(xx-yy,MS=0); 1A1 ¾¾® s2n2pp2 - s2n2s2

                                1D(yx,MS=0); 1A2 ¾¾® s2n2spp

                                1D(zx,MS=0); 1B1 ¾¾® s2n2s*pp

                                1D(zy,MS=0); 1B2 ¾¾® s2n2s*s

                                1D(2zz+xx+yy,MS=0); 1A1 ¾¾® 2s2n2s*2 + s2n2pp2 + s2n2s2

Note, the C + H2 state to which the lowest 1A1 (s2n2s2) CH2 state decomposes would be sg2s2py2.  This state (sg2s2py2) cannot be obtained by a simple combination of the 1D states.  In order to obtain pure sg2s2py2 it is necessary to combine 1S with 1D.  For example,

sg2s2py2 = - .

This indicates that a configuration correlation diagram must be drawn with a barrier near the 1D asymptote to represent the fact that 1A1 CH2 correlates with a mixture of 1D and 1S carbon plus hydrogen.  The C + H2 state to which the lowest 3B1 (s2ns2pp) CH2 state decomposes would be sg2spy2px. 

 

 

            f. If you follow the 3B1 component of the C(3P) + H2 (since it leads to the ground-state products) to 3B1 CH2 you must go over an approximately 20 Kcal/mole barrier.  Of course this path produces 3B1 CH2 product.  Distortions away from C2v symmetry, for example to Cs symmetry, would make the a1 and b2 orbitals identical in symmetry (a').  The b1 orbitals would maintain their different symmetry going to a'' symmetry.  Thus 3B1 and 3A2 (both 3A'' in Cs symmetry and odd under reflection through the molecular plane) can mix.  The system could thus follow the 3A2 component of the C(3P) + H2 surface to the place (marked with a circle on the CCD) where it crosses the 3B1 surface upon which it then moves and continues to products.  As a result, the barrier would be lowered.

            You can estimate when the barrier occurs (late or early) using thermodynamic information for the reaction (i.e. slopes and asymptotic energies).  For example, an early barrier would be obtained for a reaction with the characteristics:

 

           

 

and a late barrier would be obtained for a reaction with the characteristics:

 

           

 

This relation between reaction endothermicity or exothermicity and the character of the transition state is known as the Hammond postulate.  Note that the C(3P1) + H2 --> CH2 reaction of interest here has an early barrier.

            g. The reaction C(1D) + H2 ---> CH2 (1A1) should have no symmetry barrier (this can be recognized by following the 1A1 (C(1D) + H2) reactants down to the 1A1 (CH2) products).

 

25.

 

 

This problem in many respects is analogous to problem 24.

The 3B1 surface certainly requires a two configuration CI wavefunction; the s2s2npx (p2py2spx) and the s2n2pxs* (p2s2pxpz).  The 1A1 surface could use the s2s2n2 (p2s2py2) only but once again there is no combination of 1D determinants which gives purely this configuration (p2s2py2).  Thus mixing of both 1D and 1S determinants are necessary to yield the required p2s2py2 configuration.  Hence even the 1A1 surface would require a multiconfigurational wavefunction for adequate description.

 

           

 

 

Configuration correlation diagram for the reaction C2H2 + C ---> C3H2.

 

 

 

26.

 

a. CCl4 is tetrahedral and therefore is a spherical top.  CHCl3 has C3v symmetry and therefore is a symmetric top.  CH2Cl2 has C2v symmetry and therefore is an asymmetric top. 

            b. CCl4 has such high symmetry that it will not exhibit pure rotational spectra because it has no permanent dipole moment. CHCl3 and CH2Cl2 will both exhibit pure rotation spectra.

 

 

 

27.

 

NH3 is a symmetric top (oblate).  Use the given energy expression,

                                    E = (A - B) K2 + B J(J + 1),

A = 6.20 cm-1, B = 9.44 cm-1, selection rules DJ = ±1, and the fact that lies along the figure axis such that DK = 0, to give:

                                    DE = 2B (J + 1) = 2B, 4B, and 6B (J = 0, 1, and 2).

So, lines are at 18.88 cm-1, 37.76 cm-1, and 56.64 cm-1.

 

28.

To convert between cm-1 and energy, multiply by hc = (6.62618x10-34J sec)(2.997925x1010cm sec-1) = 1.9865x1023 J cm.

Let all quantities in cm-1 be designated with a bar,

e.g. = 1.78 cm-1. 

            a.         hc=

                        Re = ,       

                        m = = x 1.66056x10-27 kg

                                                   = 1.0824x10-26 kg.

                        hc= hc(1.78 cm-1) = 3.5359x10-23 J

                        Re =

                        Re = 1.205x10-10 m = 1.205 Å

                        De = , = = = 6.35x10-6 cm-1

                        wexe = , = = = 13.30 cm-1.

                        D= D- + , = - +

                                                                           = 66782.2 - +

                                                                           = 65843.0 cm-1 = 8.16 eV.

                        ae = +

                        = +

                        = + = 0.0175 cm-1.

                        B0 = Be - ae(1/2) , = - = 1.78 - 0.0175/2

                                                                                                  = 1.77 cm-1

                        B1 = Be - ae(3/2) , = - = 1.78 - 0.0175(1.5)

                                                                                                  = 1.75 cm-1

            b. The molecule has a dipole moment and so it should have a pure rotational spectrum.  In addition, the dipole moment should change with R and so it should have a vibration-rotation spectrum.

The first three lines correspond to J = 1 ® 0, J = 2 ® 1, J = 3 ® 2

E = we(v + 1/2) - wexe(v + 1/2)2 + BvJ(J + 1) - DeJ2(J + 1)2

DE = we - 2wexe - B0J(J + 1) + B1J(J - 1) - 4DeJ3

= - 2- J(J + 1) + J(J - 1) - 4J3

= 1885 - 2(13.3) - 1.77J(J + 1) + 1.75J(J - 1) - 4(6.35x10-6)J3

     = 1858.4 - 1.77J(J + 1) + 1.75J(J - 1) - 2.54x10-5J3

= 1854.9 cm-1

= 1851.3 cm-1

= 1847.7 cm-1

 

 

29.

 

The C2H2Cl2 molecule has a sh plane of symmetry (plane of molecule), a C2 axis (^ to the molecular plane), and inversion symmetry, this results in C2h symmetry.  Using C2h symmetry, the modes can be labeled as follows: n1, n2, n3, n4, and n5 are ag, n6 and n7 are au, n8 is bg, and n9, n10, n11, and n12 are bu.

 

 

30.

 

 

                                               

            Molecule I                                                                               Molecule II

            RCH = 1.121 Å                                                                       RCH = 1.076 Å

            ÐHCH = 104°                                                              ÐHCH = 136°

            yH = R Sin (q/2) = ±0.8834                                        yH = ±0.9976

            zH = R Cos (q/2) = -0.6902                                         zH = -0.4031

Center of Mass(COM):

            clearly, X = Y = 0,

            Z = = -0.0986                              Z = -0.0576

 

            a.         Ixx = - M(Y2 + Z2)

                        Ixy = -- MXY

Ixx = 2(1.121)2 - 14(-0.0986)2            Ixx = 2(1.076)2 - 14(-0.0576)2

      = 2.377                                                = 2.269

Iyy = 2(0.6902)2 - 14(-0.0986)2          Iyy = 2(0.4031)2 - 14(-0.0576)2

      = 0.8167                                              = 0.2786

Izz = 2(0.8834)2                                              Izz = 2(0.9976)2

     = 1.561                                                            = 1.990

Ixz = Iyz = Ixy = 0

            b. Since the moment of inertia tensor is already diagonal, the principal moments of inertia have already been determined to be

(Ia < Ib < Ic):

Iyy < Izz < Ixx                                                 Iyy < Izz < Ixx

0.8167 < 1.561 < 2.377                                   0.2786 < 1.990 < 2.269

Using the formula: A = = X

                                       A = cm-1

similarly, B = cm-1, and C = cm-1.

So,

Molecule I                                                                   Molecule II

y Þ A = 20.62                                                            y Þ A = 60.45

z Þ B = 10.79                                                z Þ B = 8.46

x Þ C = 7.08                                                  x Þ C = 7.42

            c. Averaging B + C:

B = (B + C)/2 = 8.94                                       B = (B + C)/2 = 7.94

A - B = 11.68                                                  A - B = 52.51

Using the prolate top formula:

                        E = (A - B) K2 + B J(J + 1),

Molecule I                                                                   Molecule II

E = 11.68K2 + 8.94J(J + 1)                             E = 52.51K2 + 7.94J(J + 1)

Levels: J = 0,1,2,... and K = 0,1, ... J

For a given level defined by J and K, there are MJ degeneracies given by:   (2J + 1) x

            d.

Molecule I                                                                   Molecule II

 

 

e.  Assume molecule I is CH2- and molecule II is CH2.  Then,

DE = EJj(CH2) - EJi(CH2-), where:

E(CH2) = 52.51K2 + 7.94J(J + 1), and E(CH2-) = 11.68K2 + 8.94J(J + 1)

For R-branches: Jj = Ji + 1, DK = 0:

DER = EJj(CH2) - EJi(CH2-)

            = 7.94(Ji + 1)(Ji + 1 + 1) - 8.94Ji(Ji + 1)

            = (Ji + 1){7.94(Ji + 1 + 1) - 8.94Ji}

            = (Ji + 1){(7.94- 8.94)Ji + 2(7.94)}

            = (Ji + 1){-Ji + 15.88}

For P-branches: Jj = Ji - 1, DK = 0:

DEP = EJj(CH2) - EJi(CH2-)

            = 7.94(Ji - 1)(Ji - 1 + 1) - 8.94Ji(Ji + 1)

            = Ji{7.94(Ji - 1) - 8.94(Ji + 1)}

            = Ji{(7.94- 8.94)Ji - 7.94 - 8.94}

            = Ji{-Ji - 16.88}

This indicates that the R branch lines occur at energies which grow closer and closer together as J increases (since the 15.88 - Ji term will cancel).  The P branch lines occur at energies which lie more and more negative (i.e. to the left of the origin).  So, you can predict that if molecule I is CH2- and molecule II is CH2 then the R-branch has a band head and the P-branch does not.  This is observed, therefore our assumption was correct: molecule I is CH2- and molecule II is CH2.

            f. The band head occurs when = 0.

                        = [(Ji + 1){-Ji + 15.88}] = 0

                                      = = 0

                                      = -2Ji + 14.88 = 0

                                    \ Ji = 7.44,  so J = 7 or 8.

At J = 7.44:

            DER = (J + 1){-J + 15.88}

            DER = (7.44 + 1){-7.44 + 15.88} = (8.44)(8.44) = 71.2 cm-1 above                                      the origin.

 

 

31.

 

a.

 

 

D6h

 E

2C6

2C3

C2

3C2'

3C2"

 i

2S3

2S6

sh

3sd

3sv

 

 

A1g

 1

 1

 1

 1

 1

 1

 1

 1

 1

 1

 1

 1

 

x2+y2,z2

A2g

 1

 1

 1

 1

-1

-1

 1

 1

 1

 1

-1

-1

Rz

 

B1g

 1

-1

 1

-1

 1

-1

 1

-1

 1

-1

 1

-1

 

 

B2g

 1

-1

 1

-1

-1

 1

 1

-1

 1

-1

-1

 1

 

 

E1g

 2

 1

-1

-2

 0

 0

 2

 1

-1

-2

 0

 0

Rx,Ry

(xz,yz)

E2g

 2

-1

-1

 2

 0

 0

 2

-1

-1

 2

 0

 0

 

(x2-y2,xy)

A1u

 1

 1

 1

 1

 1

 1

-1

-1

-1

-1

-1

-1

 

 

A2u

 1

 1

 1

 1

-1

-1

-1

-1

-1

-1

 1

 1

z

 

B1u

 1

-1

 1

-1

 1

-1

-1

 1

-1

 1

-1

 1

 

 

B2u

 1

-1

 1

-1

-1

 1

-1

 1

-1

 1

 1

-1

 

 

E1u

 2

 1

-1

-2

 0

 0

-2

-1

 1

 2

 0

 0

(x,y)

 

E2u

 2

-1

-1

 2

 0

 0

-2

 1

 1

-2

 0

 0

 

 

GC-H

6

0

0

0

0

2

0

0

0

6

2

0

 

 

 

            b. The number of irreducible representations may be found by using the following formula:

                        nirrep = ,

where g = the order of the point group (24 for D6h).

                        nA1g =

                                    = {(1)(6)(1)+(2)(0)(1)+(2)(0)(1)+(1)(0)(1)

                                           +(3)(0)(1)+(3)(2)(1)+(1)(0)(1)+(2)(0)(1)

                                           +(2)(0)(1)+(1)(6)(1)+(3)(2)(1)+(3)(0)(1)}

                                    = 1

                        nA2g = {(1)(6)(1)+(2)(0)(1)+(2)(0)(1)+(1)(0)(1)

                                          +(3)(0)(-1)+(3)(2)(-1)+(1)(0)(1)+(2)(0)(1)

                                          +(2)(0)(1)+(1)(6)(1)+(3)(2)(-1)+(3)(0)(-1)}

                                    = 0

                        nB1g = {(1)(6)(1)+(2)(0)(-1)+(2)(0)(1)+(1)(0)(-1)

                                          +(3)(0)(1)+(3)(2)(-1)+(1)(0)(1)+(2)(0)(-1)

                                          +(2)(0)(1)+(1)(6)(-1)+(3)(2)(1)+(3)(0)(-1)}

                                    = 0

                        nB2g = {(1)(6)(1)+(2)(0)(-1)+(2)(0)(1)+(1)(0)(-1)

                                          +(3)(0)(-1)+(3)(2)(1)+(1)(0)(1)+(2)(0)(-1)

                                          +(2)(0)(1)+(1)(6)(-1)+(3)(2)(-1)+(3)(0)(1)}

                                    = 0

                        nE1g = {(1)(6)(2)+(2)(0)(1)+(2)(0)(-1)+(1)(0)(-2)

                                          +(3)(0)(0)+(3)(2)(0)+(1)(0)(2)+(2)(0)(1)

                                          +(2)(0)(-1)+(1)(6)(-2)+(3)(2)(0)+(3)(0)(0)}

                                    = 0

                        nE2g = {(1)(6)(2)+(2)(0)(-1)+(2)(0)(-1)+(1)(0)(2)

                                          +(3)(0)(0)+(3)(2)(0)+(1)(0)(2)+(2)(0)(-1)

                                          +(2)(0)(-1)+(1)(6)(2)+(3)(2)(0)+(3)(0)(0)}

                                    = 1

                        nA1u = {(1)(6)(1)+(2)(0)(1)+(2)(0)(1)+(1)(0)(1)

                                          +(3)(0)(1)+(3)(2)(1)+(1)(0)(-1)+(2)(0)(-1)

                                          +(2)(0)(-1)+(1)(6)(-1)+(3)(2)(-1)+(3)(0)(-1)}

                                    = 0

                        nA2u = {(1)(6)(1)+(2)(0)(1)+(2)(0)(1)+(1)(0)(1)

                                          +(3)(0)(-1)+(3)(2)(-1)+(1)(0)(-1)+(2)(0)(-1)

                                          +(2)(0)(-1)+(1)(6)(-1)+(3)(2)(1)+(3)(0)(1)}

                                    = 0

                        nB1u = {(1)(6)(1)+(2)(0)(-1)+(2)(0)(1)+(1)(0)(-1)

                                          +(3)(0)(1)+(3)(2)(-1)+(1)(0)(-1)+(2)(0)(1)

                                          +(2)(0)(-1)+(1)(6)(1)+(3)(2)(-1)+(3)(0)(1)}

                                    = 0

                        nB2u = {(1)(6)(1)+(2)(0)(-1)+(2)(0)(1)+(1)(0)(-1)

                                          +(3)(0)(-1)+(3)(2)(1)+(1)(0)(-1)+(2)(0)(1)

                                          +(2)(0)(-1)+(1)(6)(1)+(3)(2)(1)+(3)(0)(-1)}

                                    = 1

                        nE1u = {(1)(6)(2)+(2)(0)(1)+(2)(0)(-1)+(1)(0)(-2)

                                          +(3)(0)(0)+(3)(2)(0)+(1)(0)(-2)+(2)(0)(-1)

                                          +(2)(0)(1)+(1)(6)(2)+(3)(2)(0)+(3)(0)(0)}

                                    = 1

                        nE2u = {(1)(6)(2)+(2)(0)(-1)+(2)(0)(-1)+(1)(0)(2)

                                          +(3)(0)(0)+(3)(2)(0)+(1)(0)(-2)+(2)(0)(1)

                                          +(2)(0)(1)+(1)(6)(-2)+(3)(2)(0)+(3)(0)(0)}

                                    = 0

We see that GC-H = A1gÅE2gÅB2uÅE1u

            c. x and y Þ E1u , z Þ A2u , so, the ground state A1g level can be excited to the degenerate E1u level by coupling through the x or y transition dipoles.  Therefore E1u is infrared active and ^ polarized.

            d. (x2 + y2, z2) Þ A1g, (xz, yz) Þ E1g, (x2 - y2, xy) Þ E2g ,so, the ground state A1g level can be excited to the degenerate E2g level by coupling through the x2 - y2 or xy transitions or be excited to the degenerate A1g level by coupling through the xz or yz transitions.  Therefore A1g and E2g are Raman active..

            e. The B2u mode is not IR or Raman active.

 

 

32.

 

a. Evaluate the z-component of mfi:

                        mfi = <2pz|e r Cosq|1s>, where y1s = e , and y2pz = r Cosq e .

                        mfi = <r Cosq e |e r Cosq|e >

                            = <r Cosq e |e r Cosq|e >

                            = Cos2q

                            = 2p

                            = 2p Cos3q\s\up15(p0

                            = 2p

                            = = = 0.7449

            b. Examine the symmetry of the integrands for <2pz| e x |1s> and <2pz| e y |1s>.  Consider reflection in the xy plane:

 

 

 

 

 

 

Function

Symmetry

2pz

-1

x

+1

1s

+1

y

+1

 

Under this operation, the integrand of <2pz| e x |1s> is (-1)(1)(1) = -1 (it is antisymmetric) and hence <2pz| e x |1s> = 0.

Similarly, under this operation the integrand of <2pz| e y |1s> is

(-1)(1)(1) = -1 (it is also antisymmetric) and hence <2pz| e y |1s> = 0.

            c.         tR = ,

                        Ei = E2pz = -Z2 

                        Ef = E1s = -Z2 

                        Ei - Ef = Z2

Making the substitutions for Ei - Ef and |mfi| in the expression for tR we obtain:

                        tR = ,

                            = ,

                            = ,

Inserting e2 = we obtain:

                        tR = =

                            = 25.6289

                            = 25,6289 x

                             = 1.595x10-9 sec x

So, for example:

 

Atom

tR

H

1.595 ns

He+

99.7 ps

Li+2

19.7 ps

Be+3

6.23 ps

Ne+9

159 fs

 

 

33.

 

a. H = H0 + lH'(t), H'(t) = Vq(t), H0jk = Ekjk, wk = Ek/

            i= Hy

            let y(r,t) = iand insert into the Schrödinger equation:

            ie-iwjtjj = ijj

            e-iwjtjj = 0

            e-iwjt = 0

            im e-iwmt = e-iwjt

So,

            m = e-i(wjm)t

Going back a few equations and multiplying from the left by jk instead of jm we obtain:

            e-iwjt = 0

            ik e-iwkt = e-iwjt

So,

            k = e-i(wjk)t

Now, let:

            cm = cm(0) + cm(1)l + cm(2)l2 + ...

            ck = ck(0) + ck(1)l + ck(2)l2 + ...

and substituting into above we obtain:

m(0) + m(1)l + m(2)l2 + ... = lH'mj e-i(wjm)t

first order:

m(0) = 0 Þ cm(0) = 1

second order:

m(1) =

(n+1)st order:

m(n) =

Similarly:

first order:

k(0) = 0 Þ ck¹m(0) = 0

second order:

k(1) =

(n+1)st order:

k(n) =

So,

m(1) = cm(0) H'mm e-i(wmm)t = H'mm

cm(1)(t) = =

and similarly,

k(1) = cm(0) H'km e-i(wmk)t = H'km e-i(wmk)t

ck(1)(t) = Vkm =

m(2) =

m(2) = H'mj e-i(wjm)t + H'mm

cm(2) = e-i(wjm)t' -

            = -

            = -

            = + t -

Similarly,

k(2) =

            = H'kj e-i(wjk)t +

                                                                         H'km e-i(wmk)t

ck(2)(t) = e-i(wjk)t'

                                                                         - e-i(wmk)t'

            =

                                    - h,-e-i(wmk

            =

                                    + h,-e-i(wmk

            =

                                    +

So, the overall amplitudes cm, and ck, to second order are:

cm(t) = 1 + + t +

             -

ck(t) = +

                         + e-i(wmk)t +

                                     

            b. The perturbation equations still hold:

m(n) = ; k(n) =

So, cm(0) = 1 and ck(0) = 0

m(1) = H'mm

cm(1) = Vmm =

k(1) = H'km e-i(wmk)t

ck(1) = Vkm =

            =

m(2) = e-i(wmj+h)t Vmj eht e-i(wjm)t +

                                                                        Vmm eht 

cm(2) = -

            = e2ht - e2ht

k(2) = e-i(wmj+h)t H'kj e-i(wjk)t +

                                                                         H'km e-i(wmk)t

ck(2) = -

                                               

            = -

Therefore, to second order:

cm(t) = 1 + +  e2ht

ck(t) =

                                                 + 

            c. In part a. the c(2)(t) grow linearly with time (for Vmm = 0) while in part b. they remain finite for h > 0.  The result in part a. is due to the sudden turning on of the field.

            d. |ck(t)|2 =

 =

 =

|ck(t)|2 = 

Now, look at the limit as h ® 0+:

|ck(t)|2 ¹ 0 when Em = Ek

limhÆ0+a d(Em-Ek)

So, the final result is the 2nd  order golden rule expression:

|ck(t)|2 = d(Em-Ek)limhÆ0+

 

 

34.

 

a.         Tnm »

evaluating <1s|V|2s> (using only the radial portions of the 1s and 2s wavefunctions since the spherical harmonics will integrate to unity) where V = (e2/r), the change in Coulomb potential when tritium becomes He:

<1s|V|2s> = e e r2dr

<1s|V|2s> =

 =

<1s|V|2s> =

<1s|V|2s> = =

Now,

            En = -, E1s = -, E2s = -, E2s - E1s =

So,

            Tnm = = = = 0.312 (for Z = 1)

            b.         jm(r) = j1s = 2e Y00

The orthogonality of the spherical harmonics results in only s-states having non-zero values for Anm.  We can then drop the Y00 (integrating this term will only result in unity) in determining the value of A1s,2s.

            yn(r) = y2s = e

Remember for j1s Z = 1 and for y2s Z = 2

            Anm = e e r2dr

            Anm = e r2dr

Anm =

We obtain:

            Anm =

            Anm =

            Anm =

            Anm = -2

The transition probability is the square of this amplitude:

            Tnm = = = 0.25 (for Z = 1).

The difference in these two results (parts a. and b.) will become negligible at large values of Z when the perturbation becomes less significant than in the case of Z = 1.

 

 

35.

 

is along Z (lab fixed), and is along z (the C-I molecule fixed bond).  The angle between Z and z is b:

                        .= emCosb = emD

So,

I = <D|.|D> = Sinbdbdgda

                                           = emSinbdbdgda.

Now use:

            DD= *,

to obtain:

                        I = em*Sinbdbdgda.

Now use:

            Sinbdbdgda = dJjdMmdKn,

to obtain:

                        I = em*dJjdMmdKn

                          = em<J'M'10|JM><JK|J'K'10>.

We use:

            <JK|J'K'10> =

and,

            <J'M'10|JM> =

to give:

I = em

  = em8p2(-i)(J'-1+M+J'-1+K)

  = em8p2(-i)(M+K)

The 3-J symbols vanish unless: K' + 0 = K and M' + 0 = M.

So,

            I = em8p2(-i)(M+K)dM'MdK'K.

            b. and vanish unless J' = J + 1, J, J - 1

            \ DJ = ±1, 0

The K quantum number can not change because the dipole moment lies along the molecule's C3 axis and the light's electric field thus can exert no torque that twists the molecule about this axis. As a result, the light can not induce transitions that excite the molecule's spinning motion about this axis.

 

 

36.

 

a. B atom: 1s22s22p1, 2P ground state L = 1, S = , gives a degeneracy ((2L+1)(2S+1)) of 6.

            O atom: 1s22s22p4, 3P ground state L = 1, S = 1, gives a degeneracy ((2L+1)(2S+1)) of 9.

The total number of states formed is then (6)(9) = 54.

            b. We need only consider the p orbitals to find the low lying molecular states:

 

                       

 

Which, in reality look like this:

 

                       

 

This is the correct ordering to give a 2S+ ground state.  The only low-lying electron configurations are 1p35s2 or 1p45s1.  These lead to 2P and 2S+ states, respectively. 

            c. The bond orders in both states are 2.5.

            d. The 2S is + but g/u symmetry cannot be specified since this is a heteronuclear molecule.

            e. Only one excited state, the 2P, is spin-allowed to radiate to the 2S+.  Consider symmetries of transition moment operators that arise in the electric dipole contributions to the transition rate  z ® S+, x,y ® P, \ the 2P ® 2S+ is electric dipole allowed via a perpendicular band.

 

            f. Since ionization will remove a bonding electron, the BO+ bond is weaker than the BO bond.

            g. The ground state BO+ is 1S+ corresponding to a 1p4 electron configuration.  An electron configuration of 1p3 5s1 leads to a 3P and a 1P state.  The 3P will be lower in energy.  A 1p2 5s2 configuration will lead to higher lying states of 3S-, 1D, and 1S+.

            h. There should be 3 bands corresponding to formation of BO+ in the 1S+, 3P, and 1P states.  Since each of these involves removing a bonding electron, the Franck-Conden integrals will be appreciable for several vibrational levels, and thus a vibrational progression should be observed.

 

 

37.

 

a. The bending (p) vibration is degenerate.

            b. H---CºN

                    Ý

                        bending fundamental

            c. H---CºN

                    Ý

                        stretching fundamental

            d. CH stretch (n3 in figure) is s, CN stretch is s, and HCN (n2 in figure) bend is p.

            e. Under z (s) light the CN stretch and the CH stretch can be excited, since y0 = s, y1 = s and z = s provides coupling.

            f. Under x,y (p) light the HCN bend can be excited, since y0 = s, y1 = p and x,y = p provides coupling.

            g. The bending vibration is active under (x,y) perpendicular polarized light.  DJ = 0, ±1 are the selection rules for ^ transitions.  The CH stretching vibration is active under (z) || polarized light.  DJ = ±1 are the selection rules for || transitions.

 

 

 

 

 

38.

 

            F fi = ei fj = h fi + fi

Let the closed shell Fock potential be written as:

            Vij = , and the 1e- component as:

            hij = fi| - Ñ2 - |fj , and the delta as:

            dij = , so that:   hij + Vij = dijei.

using: fi = , fj = , and fk = , and transforming from the MO  to AO basis we obtain:

            Vij = CmiCgkCnjCkk

                  =

                  = Vmn where,

            Vmn = Pgk, and Pgk = ,

            hij = hmn , where

            hmn = cm| - Ñ2 - |cn , and

            dij = = .

So, hij + Vij = dijej becomes:

            hmn + Vmn = ej ,

            ej - hmn - Vmn = 0 for all i,j

            CmiCnj = 0 for all i,j

Therefore,

            Cnj = 0

This is FC = SCE in the AO basis.

 

 

 

 

39.

 

The Slater Condon rule for zero (spin orbital) difference with N electrons in N spin orbitals is:

            E = = +

                    = +

                    = +

If all orbitals are doubly occupied and we carry out the spin integration we obtain:

            E = 2+ ,

where i and j now refer to orbitals (not spin-orbitals).

 

 

40.

 

 

If the occupied orbitals obey Ffk = ekfk ,  then the expression for E in problem 39 can be rewritten as.

            E = +

We recognize the closed shell Fock operator expression and rewrite this as:

            E = + =

 

 

 

41.

 

I will use the QMIC software to do this problem.  Lets just start from the beginning.  Get the starting "guess" MO coefficients on disk.  Using the program MOCOEFS it asks us for the first and second MO vectors.  We input 1, 0 for the first mo (this means that the first MO is 1.0 times the He 1s orbital plus 0.0 times the H 1s orbital; this bonding MO is more likely to be heavily weighted on the atom having the higher nuclear charge) and 0, 1 for the second.  Our beginning LCAO-MO array looks like:  and is placed on disk in a file we choose to call "mocoefs.dat".  We also put the AO integrals on disk using the program RW_INTS.  It asks for the unique one- and two- electron integrals and places a canonical list of these on disk in a file we choose to call "ao_integrals.dat".  At this point it is useful for us to step back and look at the set of equations which we wish to solve: FC = SCE.  The QMIC software does not provide us with a so-called generalized eigenvalue solver (one that contains an overlap matrix; or metric), so in order to use the diagonalization program that is provided we must transform this equation (FC = SCE) to one that looks like (F'C' = C'E).  We do that in the following manner:

Since S is symmetric and positive definite we can find an Ssuch that SS= 1, SS = S, etc.

rewrite FC = SCE by inserting unity between FC and multiplying the whole equation on the left by S.  This gives:

            SFSSC = SSCE = SCE.

Letting:           F' = SFS

                        C' = SC, and inserting these expressions above give:

                        F'C' = C'E

Note, that to get the next iteration’s MO coefficients we must calculate C from C':

C' = SC, so, multiplying through on the left by Sgives:

SC' = SSC = C

This will be the method we will use to solve our fock equations.

Find Sby using the program FUNCT_MAT (this program generates a function of a matrix).  This program will ask for the elements of the S array and write to disk a file (name of your choice ... a good name might be "shalf") containing the Sarray.  Now we are ready to begin the iterative Fock procedure.

            a. Calculate the Fock matrix, F, using program FOCK which reads in the MO coefficients from "mocoefs.dat" and the integrals from "ao_integrals.dat" and writes the resulting Fock matrix to a user specified file (a good filename to use might be something like "fock1").

            b. Calculate F' = SFSusing the program UTMATU which reads in F and Sfrom files on the disk and writes F' to a user specified file (a good filename to use might be something like "fock1p").  Diagonalize F' using the program DIAG.  This program reads in the matrix to be diagonalized from a user specified filename and writes the resulting eigenvectors to disk using a user specified filename (a good filename to use might be something like "coef1p").  You may wish to choose the option to write the eigenvalues (Fock orbital energies) to disk in order to use them at a later time in program FENERGY.  Calculate C by using. C = SC'.  This is accomplished by using the program MATXMAT which reads in two matrices to be multiplied from user specified files and writes the product to disk using a user specified filename (a good filename to use might be something like "mocoefs.dat"). 

            c. The QMIC program FENERGY calculates the total energy:

 

            2<k|h|k> + 2<kl|kl> - <kl|lk> + ,           and

 

            ek + <k|h|k> + .

This is the conclusion of one iteration of the Fock procedure ... you may continue by going back to part a. and proceeding onward.

            d. and e. Results for the successful convergence of this system using the supplied QMIC software are as follows (this data is provided to give the student assurance that they are on the right track; alternatively one could switch to the QMIC program SCF and allow that program to iteratively converge the Fock equations):

 

The one-electron AO integrals:          

 

The two-electron AO integrals:

 

   1  1  1  1    1.054700

   2  1  1  1    0.4744000

   2  1  2  1    0.5664000

   2  2  1  1    0.2469000

   2  2  2  1    0.3504000

   2  2  2  2    0.6250000

 

The "initial" MO-AO coefficients:     

 

AO overlap matrix (S):                      

 

S                

 

**************

ITERATION 1

**************

 

The charge bond order matrix:           

 

The Fock matrix (F):  

 

S F S                 

 

The eigenvalues of this matrix (Fock orbital energies) are:

 

           

 

Their corresponding eigenvectors (C' = S * C) are:

 

           

 

The "new" MO-AO coefficients (C = S * C'):

 

           

 

The one-electron MO integrals:

 

           

 

The two-electron MO integrals:

 

   1  1  1  1    0.9779331

   2  1  1  1    0.1924623

   2  1  2  1    0.5972075

   2  2  1  1    0.1170838

   2  2  2  1   -0.0007945194

   2  2  2  2    0.6157323

 

The closed shell Fock energy from formula:

 

            2<k|h|k> + 2<kl|kl> - <kl|lk> + =        -2.84219933

 

from formula:

 

            ek + <k|h|k> + =                                              -2.80060530

 

the difference is:                                                                     -0.04159403

 

**************

ITERATION 2

**************

 

The charge bond order matrix:           

 

The Fock matrix:        

 

S F S                 

 

The eigenvalues of this matrix (Fock orbital energies) are:

 

           

 

Their corresponding eigenvectors (C' = S * C) are:

 

           

 

The "new" MO-AO coefficients (C = S * C'):

 

           

 

The one-electron MO integrals:

 

           

 

 

The two-electron MO integrals:

 

   1  1  1  1    0.9626070

   2  1  1  1    0.1949828

   2  1  2  1    0.6048143

   2  2  1  1    0.1246907

   2  2  2  1    0.003694540

   2  2  2  2    0.6158437

 

The closed shell Fock energy from formula:

 

            2<k|h|k> + 2<kl|kl> - <kl|lk> + =        -2.84349298

 

from formula:

 

            ek + <k|h|k> + =                                              -2.83573675

 

the difference is:                                                                     -0.00775623

 

**************

ITERATION 3

**************

 

The charge bond order matrix:           

 

The Fock matrix:        

 

S F S                 

 

The eigenvalues of this matrix (Fock orbital energies) are:

 

           

 

Their corresponding eigenvectors (C' = S * C) are:

 

           

 

The "new" MO-AO coefficients (C = S * C'):

 

           

 

The one-electron MO integrals:

 

           

 

The two-electron MO integrals:

 

   1  1  1  1    0.9600707

   2  1  1  1    0.1953255

   2  1  2  1    0.6060572

   2  2  1  1    0.1259332

   2  2  2  1    0.004475587

   2  2  2  2    0.6158972

 

The closed shell Fock energy from formula:

 

            2<k|h|k> + 2<kl|kl> - <kl|lk> + =        -2.84353018

 

from formula:

 

            ek + <k|h|k> + =                                              -2.84225941

 

the difference is:                                                                     -0.00127077

 

**************

ITERATION 4

**************

 

The charge bond order matrix:           

 

The Fock matrix:        

 

S F S                 

 

The eigenvalues of this matrix (Fock orbital energies) are:

 

           

 

Their corresponding eigenvectors (C' = S * C) are:

 

           

 

The "new" MO-AO coefficients (C = S * C'):

 

           

 

The one-electron MO integrals:

 

           

 

The two-electron MO integrals:

 

   1  1  1  1    0.9596615

   2  1  1  1    0.1953781

   2  1  2  1    0.6062557

   2  2  1  1    0.1261321

   2  2  2  1    0.004601604

   2  2  2  2    0.6159065

 

The closed shell Fock energy from formula:

 

            2<k|h|k> + 2<kl|kl> - <kl|lk> + =        -2.84352922

 

from formula:

 

            ek + <k|h|k> + =                                              -2.84332418

 

the difference is:                                                                     -0.00020504

 

**************

ITERATION 5

**************

 

The charge bond order matrix:           

 

The Fock matrix:        

 

S F S                 

 

The eigenvalues of this matrix (Fock orbital energies) are:

 

           

 

Their corresponding eigenvectors (C' = S * C) are:

 

           

 

The "new" MO-AO coefficients (C = S * C'):

 

           

 

The one-electron MO integrals:

 

           

 

The two-electron MO integrals:

 

   1  1  1  1    0.9595956

   2  1  1  1    0.1953862

   2  1  2  1    0.6062872

   2  2  1  1    0.1261639

   2  2  2  1    0.004621811

   2  2  2  2    0.6159078

 

The closed shell Fock energy from formula:

 

            2<k|h|k> + 2<kl|kl> - <kl|lk> + =        -2.84352779

 

from formula:

 

            ek + <k|h|k> + =                                              -2.84349489

 

the difference is:                                                                     -0.00003290

 

**************

ITERATION 6

**************

 

The charge bond order matrix:           

 

The Fock matrix:        

 

S F S                 

 

The eigenvalues of this matrix (Fock orbital energies) are:

 

           

 

Their corresponding eigenvectors (C' = S * C) are:

 

           

 

The "new" MO-AO coefficients (C = S * C'):

 

           

 

The one-electron MO integrals:

 

           

 

The two-electron MO integrals:

 

   1  1  1  1    0.9595859

   2  1  1  1    0.1953878

   2  1  2  1    0.6062925

   2  2  1  1    0.1261690

   2  2  2  1    0.004625196

   2  2  2  2    0.6159083

 

The closed shell Fock energy from formula:

 

            2<k|h|k> + 2<kl|kl> - <kl|lk> + =        -2.84352827

 

from formula:

 

            ek + <k|h|k> + =                                              -2.84352398

 

the difference is:                                                                     -0.00000429

 

**************

ITERATION 7

**************

 

The charge bond order matrix:           

 

The Fock matrix:        

 

S F S                 

 

The eigenvalues of this matrix (Fock orbital energies) are:

 

           

 

Their corresponding eigenvectors (C' = S * C) are:

 

           

 

The "new" MO-AO coefficients (C = S * C'):

 

           

 

The one-electron MO integrals:

 

           

 

The two-electron MO integrals:

 

   1  1  1  1    0.9595849

   2  1  1  1    0.1953881

   2  1  2  1    0.6062936

   2  2  1  1    0.1261697

   2  2  2  1    0.004625696

   2  2  2  2    0.6159083

 

The closed shell Fock energy from formula:

 

            2<k|h|k> + 2<kl|kl> - <kl|lk> + =        -2.84352922

 

from formula:

 

            ek + <k|h|k> + =                                              -2.84352827

 

the difference is:                                                                     -0.00000095

 

**************

ITERATION 8

**************

 

The charge bond order matrix:           

 

The Fock matrix:        

 

S F S                 

 

The eigenvalues of this matrix (Fock orbital energies) are:

 

           

 

Their corresponding eigenvectors (C' = S * C) are:

 

           

 

The "new" MO-AO coefficients (C = S * C'):

 

           

 

The one-electron MO integrals:

 

           

 

The two-electron MO integrals:

 

   1  1  1  1    0.9595841

   2  1  1  1    0.1953881

   2  1  2  1    0.6062934

   2  2  1  1    0.1261700

   2  2  2  1    0.004625901

   2  2  2  2    0.6159081

 

The closed shell Fock energy from formula:

 

            2<k|h|k> + 2<kl|kl> - <kl|lk> + =        -2.84352827

 

from formula:

 

            ek + <k|h|k> + =                                              -2.84352827

 

the difference is:                                                                     0.00000000

 

f. In looking at the energy convergence we see the following:

 

Iter

Formula 1

Formula 2

1

-2.84219933

-2.80060530

2

-2.84349298

-2.83573675

3

-2.84353018

-2.84225941

4

-2.84352922

-2.84332418

5

-2.84352779

-2.84349489

6

-2.84352827

-2.84352398

7

-2.84352922

-2.84352827

8

-2.84352827

-2.84352827

 

 

If you look at the energy differences (SCF at iteration n - SCF converged) and plot this data versus iteration number, and do a 5th order polynomial fit, we see the following:

 

In looking at the polynomial fit we see that the convergence is primarily linear since the coefficient of the linear term is much larger than those of the cubic and higher terms.

            g. The converged SCF total energy calculated using the result of problem 40 is an upper bound to the ground state energy, but, during the iterative procedure it is not.  Only at convergence does the expectation value of the Hamiltonian for the Hartree Fock determinant become equal to that given by the equation in problem 40.

            h. Yes, the 1s2 configuration does dissociate properly because at at R®¥ the lowest energy state is He + H+, which also has a 1s2 orbital occupancy (i.e., 1s2 on He and 1s0 on H+).

 

 

42.

2. At convergence the MO coefficients are:

 

            f1 =                            f2 =

and the integrals in this MO basis are:

h11 = -2.615842                      h21 = -0.1953882        h22 = -1.315354

g1111 = 0.9595841      g2111 = 0.1953881      g2121 = 0.6062934

g2211 = 0.1261700      g2221 = 004625901     g2222 = 0.6159081

            a. H = =

             =

             =

            b. The eigenvalues are E1 = -4.279131 and E2 = -2.007770.  The corresponding eigenvectors are:

            C1 = ,            C2 =

 

            c.

 =

 =

 = a- b.

            d. The third configuration |1s2s| = ,

Adding this configuration to the previous 2x2 CI results in the following 3x3 'full' CI:

            H =

 =

 

 

 

Evaluating the new matrix elements:

          H13 = H31 = *(-0.1953882 + 0.1953881) = 0.0

            H23 = H32 = *(-0.1953882 + 0.004626) = -0.269778

            H33 = -2.615842 - 1.315354 + 0.606293 + 0.126170

                  = -3.198733

 =

            e. The eigenvalues are E1 = -4.279345, E2 = -3.256612 and E3 = -1.949678.  The corresponding eigenvectors are:

C1 = , C2 = , C3 =

            f. We need the non-vanishing matrix elements of the dipole operator in the MO basis.  These can be obtained by calculating them by hand.  They are more easily obtained by using the TRANS program.  Put the 1e- AO integrals  on disk by running the program RW_INTS.  In this case you are inserting z11 = 0.0, z21 = 0.2854, and z22 = 1.4 (insert 0.0 for all the 2e- integrals) ... call the output file "ao_dipole.ints" for example.  The converged MO-AO coefficients should be in a file ("mocoefs.dat" is fine).  The transformed integrals can be written to a file (name of your choice) for example  "mo_dipole.ints".  These matrix elements are:

            z11 = 0.11652690, z21 = -0.54420990, z22 = 1.49117320

The excitation energies are E2 - E1 = -3.256612 - -4.279345 = 1.022733, and

E3 - E1 = -1.949678.- -4.279345 = 2.329667.

Using the Slater-Conden rules to obtain the matrix elements between configurations we obtain:

            Hz =

                =

                =

Now, <Y1|z|Y2> = C1THzC2, (this can be accomplished with the program UTMATU)

 = T 

 = -.757494

and, <Y1|z|Y3> = C1THzC3

 = T 

 = 0.014322

            g. Using the converged coefficients the orbital energies obtained from solving the Fock equations are e1 = -1.656258 and e2 = -0.228938.  The resulting expression for the PT first-order wavefunction becomes:

        |1s2>(1) = - |2s2>

        |1s2>(1) = - |2s2>

        |1s2>(1) = -0.0441982|2s2>

            h. As you can see from part c., the matrix element <1s2|H|1s2s> = 0 (this is also a result of the Brillouin theorem) and hence this configuration does not enter into the first-order wavefunction.

            i. |0> = |1s2> - 0.0441982|2s2>.  To normalize we divide by:

            = 1.0009762

        |0> = 0.999025|1s2> - 0.044155|2s2>

In the 2x2 CI we obtained:

        |0> = 0.99845123|1s2> - 0.05563439|2s2>

            j. The expression for the 2nd order RSPT is:

            E(2) = - = -

                        = -0.005576 au

Comparing the 2x2 CI energy obtained to the SCF result we have:

-4.279131 - (-4.272102) = -0.007029 au

 

43.       STO total energy:        -2.8435283

            STO3G total energy    -2.8340561

            3-21G total energy      -2.8864405

The STO3G orbitals were generated as a best fit of 3 primitive Gaussians (giving 1 CGTO) to the STO.  So, STO3G can at best reproduce the STO result.  The 3-21G orbitals  are more flexible since there are 2 CGTOs per atom.  This gives 4 orbitals  (more parameters to optimize) and a lower total energy.

44.

 

R

HeH+ Energy

H2 Energy

1.0

-2.812787056

-1.071953297

1.2

-2.870357513

-1.113775015

1.4

-2.886440516

-1.122933507

1.6

-2.886063576

-1.115567684

1.8

-2.880080938

-1.099872589

2.0

-2.872805595

-1.080269098

2.5

-2.856760263

-1.026927710

10.0

-2.835679293

-0.7361705303

 

Plotting total energy vs. geometry for HeH+:

 

 

 

Plotting total energy  vs. geometry for H2:

 

 

For HeH+ at R = 10.0 au, the eigenvalues of the converged Fock matrix and the corresponding converged MO-AO coefficients are:

 

-.1003571E+01

-.4961988E+00

 .5864846E+00

 .1981702E+01

 .4579189E+00

-.8245406E-05

 .1532163E-04 

 .1157140E+01

 .6572777E+00

-.4580946E-05

-.6822942E-05

-.1056716E+01

-.1415438E-05

 .3734069E+00

 .1255539E+01 

-.1669342E-04

 .1112778E-04

 .7173244E+00

-.1096019E+01 

 .2031348E-04

 

Notice that this indicates that orbital 1 is a combination of the s functions on He only (dissociating properly to He + H+).

 

For H2 at R = 10.0 au, the eigenvalues of the converged Fock matrix and the corresponding converged MO-AO coefficients are:

 

-.2458041E+00

-.1456223E+00

 .1137235E+01

 .1137825E+01

 .1977649E+00

-.1978204E+00

 .1006458E+01

-.7903225E+00

 .5632566E+00

-.5628273E+00

-.8179120E+00

 .6424941E+00

 .1976312E+00

 .1979216E+00

 .7902887E+00

 .1006491E+01

 .5629326E+00

 .5631776E+00

-.6421731E+00

-.8181460E+00

 

Notice that this indicates that orbital 1 is a combination of the s functions on both H atoms (dissociating improperly; equal probabilities of H2 dissociating to two neutral atoms or to a proton plus hydride ion).

 

45. The H2 CI result:

 

R

1Sg+

3Su+

1Su+

1Sg+

  1.0

-1.074970

-0.5323429

-0.3997412

 0.3841676

  1.2

-1.118442

-0.6450778

-0.4898805

 0.1763018

  1.4

-1.129904

-0.7221781

-0.5440346

 0.0151913

  1.6

-1.125582

-0.7787328

-0.5784428

-0.1140074

  1.8

-1.113702

-0.8221166

-0.6013855

-0.2190144

  2.0

-1.098676

-0.8562555

-0.6172761

-0.3044956

  2.5

-1.060052

-0.9141968

-0.6384557

-0.4530645

  5.0

-0.9835886

-0.9790545

-0.5879662

-0.5802447

  7.5

-0.9806238

-0.9805795

-0.5247415

-0.5246646

10.0

-0.980598

-0.9805982

-0.4914058

-0.4913532

 

For H2 at R = 1.4 au, the eigenvalues of the Hamiltonian matrix and the corresponding determinant amplitudes are:

 

 

 

determinant

-1.129904

-0.722178

-0.544035

0.015191

|1sga1sgb|

 0.99695

 0.00000

 0.00000

 0.07802

|1sgb1sua|

 0.00000

 0.70711

 0.70711

 0.00000

|1sga1sub|

 0.00000

 0.70711

-0.70711

 0.00000

|1sua1sub|

-0.07802

 0.00000

 0.00000

 0.99695

 

This shows, as expected, the mixing of the first 1Sg+ (1sg2) and the 2nd 1Sg+ (1su2) determinants in the first and fourth states, and the

            3Su+ = (),

and  1Su+=  ()

states as the second and third states.

 

Also notice that the first 1Sg+ state has coefficients (0.99695 - 0.07802) (note specifically the + - combination) and the second 1Sg+ state has the opposite coefficients with the same signs (note specifically the + + combination).  The + + combination always gives a higher energy than the + - combination.

 

 

46.

F atoms have 1s22s22p5   2P ground electronic states that are split by spin-orbit coupling into 2P3/2 and 2P1/2 states that differ by only 0.05 eV in energy.

 

a.

The degeneracy of a state having a given J is 2J+1, and the J=3/2 state is lower in energy because the 2p orbital shell is more than half filled (I learned this in inorganic chemistry class), so

 

qel = 4 exp(-0/kT) + 2 exp(-0.05 eV/kT).

 

0.05 eV is equivalent to k(500 K), so 0.05/kT = 500/T, hence

 

qel = 4 exp(-0/kT) + 2 exp(-500/T).

 

 

b.

Q = qN/N!

 

so, ln Q = N lnq – lnN!

 

E =kT2  ∂lnQ/∂T = NkT2 ∂lnq/∂T = Nk{1000 exp(-500/T)/[4 + 2 exp(-500/T)]}

c. Using the fact that kT=0.03eV at T=300°K, make a (qualitative) graph of /N vs T for T ranging from 100°K to 3000°K.

 

 

At T = 100 K, E/N is small and equal to 1000k exp(-5)/(4 + 2 exp(-5)).

At T = 3000 K, E/N has grown to 1000k exp(-1/6)/(4 + 2 exp(-1/6)) which is

approximately 1000k/6.

 

 

 

47.

a.

 

The difference between a linear and bent transition state would arise in the vibrational and rotational partition functions. For the linear TS, one has 3N-6 vibrations (recall that one loses one vibration as a reaction coordinate), but for the bent TS, one has 3N-7 vibrations. For the linear TS, one has 2 rotational axes, and for the bent TS, one has 3.

So the ratio of rate constants will reduce to ratios of vibration and rotation partition functions. In particular, one will have

 

klinear/kbent = (qvib3N-6 qrot2/qvib3N-7qrot3) = (qvib/qrot).

 

b. Using

 

            qt ~ 108, qr ~ 102, qv ~ 1,

 

I would expect klinear/kbent to be of the order of 1/102 = 10-2.

 

 

 

48.

 

Constructing the Slater determinant corresponding to the "state" 1s(a)1s(a) with the rows labeling the orbitals and the columns labeling the electron gives:

            |1sa1sa| =

                             =

                             = 0

 

49.

 

Starting with the MS=1 3S state (which in a "box" for this ML=0, MS=1 case would contain only one product function; |1sa2sa|) and applying S- gives:

            S- 3S(S=1,MS=1)        = 3S(S=1,MS=0)

                                                = 3S(S=1,MS=0)

 

                                                = |1sa2sa|

                                                = S-(1)|1sa2sa| + S-(2)|1sa2sa|

                                                = |1sb2sa|

                                                 + |1sa2sb|

                                                =

So,       3S(S=1,MS=0) =

                   3S(S=1,MS=0) =

The three triplet states are then:

                                3S(S=1,MS=1)= |1sa2sa|,

                                3S(S=1,MS=0) = , and

                                3S(S=1,MS=-1) = |1sb2sb|.

The singlet state which must be constructed orthogonal to the three singlet states (and in particular to the 3S(S=1,MS=0) state) can be seen to be:

                                1S(S=0,MS=0) = .

Applying S2 and Sz to each of these states gives:

                                Sz |1sa2sa|      = |1sa2sa|

                                                        = Sz(1)|1sa2sa| + Sz(2))|1sa2sa|

                                                = |1sa2sa| + |1sa2sa|

                                                = |1sa2sa|

                                S2 |1sa2sa|      = (S-S+ + Sz2 + Sz) |1sa2sa|

                                                = S-S+|1sa2sa| + Sz2|1sa2sa| + Sz|1sa2sa|

                                                = 0 + 2 |1sa2sa| + 2|1sa2sa|

                                                = 22 |1sa2sa|

Sz =

                                                            = |1sb2sa|

                                                            + |1sa2sb|

                                                            = |1sb2sa|

                                                            + |1sa2sb|

                                                            = 0

S2 = (S-S+ + Sz2 + Sz)

                        = S-S+ 

                        =

                        =

                        = 2

                        = 2

                        = 2 2 

                                Sz |1sb2sb|      = |1sb2sb|

                                                        = Sz(1)|1sb2sb| + Sz(2))|1sb2sb|

                                                = |1sb2sb| + |1sb2sb|

                                                = -|1sb2sb|

                                S2 |1sb2sb|      = (S+S- + Sz2 - Sz) |1sb2sb|

                                                = S+S-|1sb2sb| + Sz2|1sb2sb| - Sz|1sb2sb|

                                                = 0 + 2 |1sb2sb| + 2|1sb2sb|

                                                = 22 |1sb2sb|

Sz =

                                                            = |1sb2sa|

                                                            - |1sa2sb|

                                                            = |1sb2sa|

                                                            - |1sa2sb|

                                                            = 0

S2 = (S-S+ + Sz2 + Sz)

                        = S-S+ 

                        =

                        =

                        = 0

                        = 0

                        = 0 2 

 

 

 

50.

 

As shown in problem 22c, for two equivalent p electrons one obtains six states:

            1D (ML=2); one state (MS=0),

            1D (ML=-2); one state (MS=0),

            1S (ML=0); one state (MS=0), and

            3S (ML=0); three states (MS=1,0, and -1).

By inspecting the "box" in problem 22c, it should be fairly straightforward to write down the wavefunctions for each of these:

            1D (ML=2); |p1ap1b|

            1D (ML=-2); |p-1ap-1b|

            1S (ML=0);

            3S (ML=0, MS=1); |p1ap-1a|

            3S (ML=0, MS=0);

            3S (ML=0, MS=-1); |p1bp-1b|

 

 

51.

 

We can conveniently couple another s electron to the states generated from the 1s12s1 configuration:

                3S(L=0, S=1) with 3s1(L=0, S=) giving:

            L=0, S=, ; 4S (4 states) and 2S (2 states).

                1S(L=0, S=0) with 3s1(L=0, S=) giving:

            L=0, S=; 2S (2 states).

Constructing a "box" for this case would yield:

 

 

 

 

 

 

 

 

                                              ML

MS

0

|1sa2sa3sa|

|1sa2sa3sb|,    |1sa2sb3sa|, |1sb2sa3sa|

 

One can immediately identify the wavefunctions for two of the quartets (they are single entries):

            4S(S=,MS=): |1sa2sa3sa|

            4S(S=,MS=-): |1sb2sb3sb|

Applying S- to 4S(S=,MS=) yields:

            S-4S(S=,MS=) = 4S(S=,MS=)

                                          = 4S(S=,MS=)

            S-|1sa2sa3sa| =

So, 4S(S=,MS=) =

Applying S+ to 4S(S=,MS=-) yields:

            S+4S(S=,MS=-) = 4S(S=,MS=-)

                                          = 4S(S=,MS=-)

            S+|1sb2sb3sb| =

So, 4S(S=,MS=-) =

It only remains to construct the doublet states which are orthogonal to these quartet states.  Recall that the orthogonal combinations for systems having three equal components (for example when symmetry adapting the 3 sp2 hybrids in C2v or D3h symmetry) give results of + + +, +2 - -, and 0 + -.  Notice that the quartets are the + + + combinations and therefore the doublets can be recognized as:

2S(S=,MS=) =

2S(S=,MS=) =

2S(S=,MS=-) =

2S(S=,MS=-) =

 

52.

 

As illustrated in problem 24, a p2 configuration (two equivalent p electrons) gives rise to the term symbols: 3P, 1D, and 1S.  Coupling an additional electron (3d1) to this p2 configuration will give the desired 1s22s22p23d1 term symbols:

            3P(L=1,S=1) with 2D(L=2,S=) generates;

            L=3,2,1, and S=, with term symbols 4F, 2F,4D, 2D,4P, and 2P,

            1D(L=2,S=0) with 2D(L=2,S=) generates;

            L=4,3,2,1,0, and S=with term symbols 2G, 2F, 2D, 2P, and 2S,

            1S(L=0,S=0) with 2D(L=2,S=) generates;

            L=2 and S=with term symbol 2D.

 

 

53. The notation used for the Slater Condon rules will be as follows:

            (a.) zero (spin orbital) difference;

                        = +

                                            = +

            (b.) one (spin orbital) difference (fp ¹ fp');

                        = +

                                            = fpp' +

            (c.) two (spin orbital) differences (fp ¹ fp' and fq ¹ fq');

                        = -

                                            = gpqp'q' - gpqq'p'

            (d.) three or more (spin orbital) differences;

                        = 0

 

 

i. 3P(ML=1,MS=1) = |p1ap0a|

= <| 10| H | 10|>

 

  Using the Slater Condon rule (a.) above (I will denote these SCa-SCd):

                        = f11 + f00 + g1010 - g1001

 

ii. 3P(ML=0,MS=0) =

                       

                         =

                             + + )

Evaluating each matrix element gives:

= f1a1a + f-1b-1b + g1a-1b1a-1b - g1a-1b-1b1a (SCa)

                                                            = f11 + f-1-1 + g1-11-1 - 0

= g1a-1b1b-1a - g1a-1b-1a1b (SCc)

                                                            = 0 - g1-1-11

= g1b-1a1a-1b - g1b-1a-1b1a (SCc)

                                                            = 0 - g1-1-11 

= f1b1b + f-1a-1a + g1b-1a1b-1a - g1b-1a-1a1b (SCa)

                                                            = f11 + f-1-1 + g1-11-1 - 0

Substitution of these expressions give:

                       

                         =

                                                 + f11 + f-1-1 + g1-11-1)

                         = f11 + f-1-1 + g1-11-1 - g1-1-11

 

iii. 1S(ML=0,MS=0);

                       

                         =

                             - -

                             + +

                             - +

                             + )

Evaluating each matrix element gives:

= f0a0a + f0b0b + g0a0b0a0b - g0a0b0b0a (SCa)

                                                = f00 + f00 + g0000 - 0

=

                                                = g0a0b1a-1b - g0a0b-1b1a (SCc)

                                                = g001-1 - 0

=

                                                = g0a0b-1a1b - g0a0b1b-1a (SCc)

                                                = g00-11 - 0

= f1a1a + f-1b-1b + g1a-1b1a-1b - g1a-1b-1b1a (SCa)

                                                = f11 + f-1-1 + g1-11-1 - 0

=

                                                = g1a-1b-1a1b - g1a-1b1b-1a (SCc)

                                                = g1-1-11 - 0

= f-1a-1a + f1b1b + g-1a1b-1a1b - g-1a1b1b-1a (SCa)

                                                = f-1-1 + f11 + g-11-11 - 0

Substitution of these expressions give:

           

             =

                                 + g1-11-1 + g1-1-11 - g00-11 + g1-1-11 + f-1-1 + f11 + g-11-11)

             =

 

iv. 1D(ML=0,MS=0) =

Evaluating we note that all the Slater Condon matrix elements generated are the same as those evaluated in part iii. (the signs for the wavefunction components and the multiplicative factor of two for one of the components, however, are different).

           

             =

                         + f-1-1 + g1-11-1 + g1-1-11 + 2g00-11 + g1-1-11 + f-1-1 + f11

                                 + g-11-11)

             =

 

54.

 

 

i. 1D(ML=2,MS=0) = |p1ap1b|

                       

                         =

                         = f1a1a + f1b1b + g1a1b1a1b - g1a1b1b1a (SCa)

                         = f11 + f11 + g1111 - 0

                         = 2f11 + g1111

 

ii. 1S(ML=0,MS=0) =

                       

                         =

                             - + )

Evaluating each matrix element gives:

= f1a1a + f-1b-1b + g1a-1b1a-1b - g1a-1b-1b1a (SCa)

                                                = f11 + f-1-1 + g1-11-1 - 0

= g1a-1b1b-1a - g1a-1b-1a1b (SCc)

                                                = 0 - g1-1-11

= g1b-1a1a-1b - g1b-1a-1b1a (SCc)

                                                = 0 - g1-1-11 

= f1b1b + f-1a-1a + g1b-1a1b-1a - g1b-1a-1a1b (SCa)

                                                = f11 + f-1-1 + g1-11-1 - 0

Substitution of these expressions give:

           

             =

             = f11 + f-1-1 + g1-11-1+ g1-1-11

 

iii. 3S(ML=0,MS=0) =

                       

                         

                                                = f11 + f-1-1 + g1-11-1 - 0

= g1a-1b1b-1a - g1a-1b-1a1b (SCc)

                                                = 0 - g1-1-11

= g1b-1a1a-1b - g1b-1a-1b1a (SCc)

                                                = 0 - g1-1-11 

= f1b1b + f-1a-1a + g1b-1a1b-1a - g1b-1a-1a1b (SCa)

                                                = f11 + f-1-1 + g1-11-1 - 0

Substitution of these expressions give:

           

             =

             = f11 + f-1-1 + g1-11-1- g1-1-11

 

 

55.

 

The order of the answers is J, I, G. K, B, D, E, A, C, H, F

 

 

56.

 

p = N/(V-Nb) – N2 a/(kTV2)

 

but p/kT = (∂lnQ/∂V)T,N

 

so we can integrate to obtain ln Q

 

lnQ = ò (p/kT) dV = ò [N/(V-Nb) – N2 a/(kTV2)] dV

 

= N ln(V-Nb) + N2a/kT (1/V)

 

So,

 

Q = {(V-Nb)exp[(a/kT) (N/V)]}N

 

57.

a.

MD because you need to keep track of how far the molecule moves as a function of time and MC does not deal with time.

 

b.

 

MC is capable of doing this although MD is also. However, MC requires fewer computational steps, so I would prefer to use it.

 

c.

 

MC can do this, as could MD. Again, because MC needs fewer computational steps, I’d use it.

 

 

            Suppose you are carrying out a Monte-Carlo simulation involving 1000 Ar atoms.  Further suppose that the potentials are pairwise additive and that your computer requires approximately 50 floating point operations (FPO's) (e.g. multiply, add, divide, etc.) to compute the interaction potential between any pair of atoms

 

 

d.

For each MC move, we must compute only the change in potential energy. To do this, we need to compute only the change in the pair energies that involve the atom that was moved. This will require 999x50 FPOs (the 99 being  the number of atoms other than the one that moved). So, for a million MC steps, I would need 106 x 999 x 50 FPOs. At 100 x106 FPOs per second, this will require 495 seconds, or a little over eight minutes.

 

e.

Because the statistical fluctuations in MC calculations are proportional to (1/N)1/2, where N is the number of steps taken, I will have to take 4 times as many steps to cut the statistical errors in half. So, this will require 4 x 495 seconds or 1980 seconds.

 

 

f.

 

If we have one million rather than one thousand atoms, the 495 second calculation of part d would require

 

999,999/999

 

times as much time. This ratio arises because the time to compute the change in potential energy accompanying a MC move is proportional to the number of other atoms. So, the calculation would take 495 x (999,999/999) seconds or about 500,000 seconds or about 140 hours.

 

g.

 

We would be taking 10-9s/(10-15 s per step) = 106 MD steps.

Each step requires that we compute all forces(-∂V∂RI,J) between all pairs of atoms. There are 1000x999/2 such pairs. So, to compute all the forces would require

(1000x999/2)x 50 FPOs = 2.5 x107 FPOs. So, we will need

 

2.5 x107 FPOs/step x 106 steps/(100 FPOs per second)

= 2.5 x105 seconds or about 70 hours.

 

h.

 

The graduate student is 108 times slower than the 100 Mflop computer, so it will take her/him 108 times as long, so 495 x108 seconds or about 1570 years.

 

58.

 

First, Na has a 2S ground state term symbol whose degeneracy is 2S + 1 = 2.

Na2 has a 1S ground state whose degeneracy is 1.

The symmetry number for Na2 is s = 2.

The D0 value given is 17.3 kcal mol-1.

 

The Kp equilibrium constant would be given in terms of partial pressures as (and then using pV=NkT)

 

Kp = pNa2/pNa2 = (kT)-1 (qNa/V)2/(qNa2/V)

 

in terms of the partition functions.

 

a.

qNa = (2pmkT/h2)3/2 V q­el

 

qNA2 = (2pm’kT/h2)3/2 V (8p2IkT/h2) 1/2 [ exp-hn/2kT) (1- exp-hn/kT))-1 exp(De/kT)

 

We can combine the De and the –hn/2kT to obtain the D0 which is what we were given.

 

b. For Na (I will use cgs units in all cases):

 

q/V = (2p 23 1.66x10-24 1.38 x10-16 1000)3/2 2

= (6.54 x1026) x 2 = 1.31 x1027

 

For Na2:

 

q/N = 23/2 x (6.54 x1026) (1000/0.221) (1/2) (1-exp(-229/1000))-1 exp(D0/kT)

= 1.85 x1027  (2.26 x103) (4.88) (5.96 x103)

= 1.22 x1035

 

So,

 

Kp = [1.22 x1035]/[(1.38 x10-16)(1000) (1.72 x1054)

= 0.50 x10-6 dynes cm-2 = 0.50 atm-1.

 

 

59.

 

The differences in krate will arise from differences in the number of translational, rotational, and vibrational partition functions arising in the adsorbed and gas-phase species. Recall that

 

krate = (kT/h) exp(-E*/kT) [qTS/V]/[(qNO/V) (qCl2/V)]

 

In the gas phase,

 

NO has 3 translations, two rotations, and one vibration
Cl2 has 3 translations, two rotations, and one vibration

the NOCl2 TS, which is bent, has 3 translations, three rotations, and five vibrations (recall that one vibration is missing and is the reaction coordinate)

 

In the adsorbed state,

 

NO has 2 translations, one rotation, and three vibrations
Cl2 has 2 translations, one rotation, and three vibrations

the NOCl2 TS, which is bent, has 2 translations, one rotation, and eight vibrations (again, one vibration is missing and is the reaction coordinate).

 

So, in computing the partition function ratio:

 

[qTS/V]/[(qNO/V) (qCl2/V)]

 

for the adsorbed and gas-phase cases, one does not obtain the same number of translational, rotational, and vibrational factors. In particular, the ratio of these factors for the adsorbed and gas-phase cases gives the ratio of rate constants as follows:

 

kad/kgas = (qtrans/V)/qvib

 

which should be of the order of 108 (using the ratio of partition functions as given).

Notice that this result suggests that reaction rates can be altered by constraining the reacting species to move freely in lower dimensions even if one does not alter the energetics (e.g., activation energy or thermochemistry). 

Contributions

Jack Simons (Henry Eyring Scientist and Professor of Chemistry, U. Utah) Telluride Schools on Theoretical Chemistry

Integrated by Tomoyuki Hayashi (UC Davis)