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Workgroup 1: Valence Bond Orbitals for Hydrogen

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    52954
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    Section: _____________________________

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    Separate into groups of five-eight students and answer the following questions. You will be graded primarily on effort, not accuracy (although if all questions are incorrect, then the effort is clearly missing). It is OK to use external resources such as textbooks and websites. It is not ok to copy word-for-word the solution. Each group must come up with their answers collectively and without plagiarism of existing resources.

    A quantum description of bonding in molecules requires generating the wavefunctions that are solutions to the multi-body (involving both nuclei and electrons) Schrödinger equation. These wavefunctions must describe each electron as accurately as quantum mechanics can. The two theories commonly used in constructing these wavefunctions are valence bond (VB) theory and molecular orbital (MO) theory. This activity addresses a topical application of these theories to diatomic hydrogen.

    For hydrogen molecule \(H_2\), write out the Schrödinger equation that needs to be solved? Label each term in the Hamiltonian.

     

     

     

     

     

     

     

     

    When one "solves a Schrödinger equation", what do you expect to obtain? How would you analytically solve the Schrödinger equation written above?

     

     

     

     

     

     

     

     

    Valence Bond theory is the first quantum mechanical theory of bonding and only considers the bonding between two atoms at a time as “localized bonds”. Essential features of VB theory is that an electron pair between atoms forms a (covalent) bond. Other electron pairs are lone pairs. All bonds are constructed by the overlap of atomic orbitals on the constituent atoms. For the following discussion, we consider the orbital on hydrogen atoms A and B. Let's represent electron 1 on the 1s atomic orbital of hydrogen A as \(|1S_A(1) \rangle\). Likewise, we represent electron 2 on the 1s atomic orbital of hydrogen B as \(|1S_B(2) \rangle\).

    How would you interpret (i.e. describe) the following two-electron wavefunction: \(| \psi_1(1,2) \rangle = |1S_A(1) \rangle |1S_B(2) \rangle\)? Is this an expected bonding description (i.e., describe the molecule)? Does this wavefunction better describe Schrödinger equation when the two hydrogen nuclei are very far apart?  Plot the amplitude of this wavefunction as a slice along the \(H_2\) axis.

     

     

     

    How would you interpret (i.e. describe) the following two-electron wavefunction: \(| \psi_2(1,2) \rangle = |1S_A(2) \rangle |1S_B(1) \rangle\)? Is this an expected bonding description (i.e., describe the molecule); why or why not? Can it describe the situation when the two hydrogen atoms are very far apart?  Plot the amplitude of this wavefunction as a slice along the \(H_2\) axis.

     

     

     

     

    Are \(| \psi_1(1,2) \rangle\) and  \(| \psi_2(1,2) \rangle\) normalized wavefunctions? If not, what are the corresponding normalized wavefunctions?

     

     

     

     

     

     

     

    At great distances, it is possible to distinguish whether electron 1 is on atom A and electron whether electron 1 is on atom A and electron 2 is on atom B. 2 is on atom B. At close distances, however, it is not possible At close distances, however, it is not possible to distinguish which atom electron 1 or to distinguish which atom electron 1 or electron 2 are on. Because of this, \(|1S_A(1) \rangle \) and \(|1S_A(1) \rangle \) are equally valid descriptions.That means we can construct a linear combination of them,

    How would you interpret (i.e. describe) the following two-electron wavefunction:

    \[| \psi_3(1,2) \rangle  =  |1S_A(1) \rangle |1S_B(2) \rangle  + |1S_A(2) \rangle |1S_B(1) \rangle\]

    Is this an expected bonding description (i.e., describe the molecule); why or why not? Plot the amplitude of this wavefunction as a slice along the \(H_2\) axis.

     

     

     

     

     

     

     

     

     

     

     

    How would you interpret (i.e. describe) the following two-electron wavefunction:

    \[| \psi_4(1,2) \rangle = |1S_A(1) \rangle |1S_B(2) \rangle  - |1S_A(2) \rangle |1S_B(1) \rangle\]

    Is this an expected bonding description (i.e., describe the molecule); why or why not? Plot the amplitude of this wavefunction as a slice along the \(H_2\) axis.

     

     

     

     

     

    Are \(| \psi_3(1,2) \rangle\) and  \(| \psi_4(1,2) \rangle\) normalized wavefunctions? If not, what are the corresponding normalized wavefunctions?

     

     

     

     

     

     

     

     

    \( | \psi_3(1,2) \rangle \) is the bonding orbital and \( |\psi_4(1,2) \rangle \) is the antibonding orbital for \(H_2\) within valence bond theory. Which is higher in energy?

     

     

     

     

     

     

     

     

     

     

     

     

     

     

     

    Within VB theory, each electron pair is described by a wavefunction formed of the product (or sum of products) of the atomic orbital wave functions (e.g., \( | \psi_3(1,2) \rangle \)  and \( | \psi_4(1,2) \rangle \). Molecular orbital theory predicts different wavefunctions as you remember (or have to lookup):

    \[| \psi_5 (1,2) \rangle = | \sigma_{1s} \rangle \]

    and

    \[ | \psi_6 (1,2) \rangle = | \sigma_{1s}^* \rangle\]

    What are the expressions of these MO wavefunctions in terms of that atomic orbitals of the hydrogen atoms.

     

     

     

     

     

     

     

    Plot the amplitude of bonding ( \( | \psi_3 (1,2) \rangle \)  vs. \( | \psi_5 (1,2) \rangle \)) and antibonding  ( \( | \psi_4 (1,2) \rangle \)  vs. \( | \psi_6 (1,2) \rangle \)) wavefunctions along a slice along the \(H_2\) axis. How do they differ? Do you expect the two theories to predict similar energies for the bonding and anitbonding orbitals?

     

     

     

     

     

    Neither theory is perfect. If the hydrogen nuclei are far apart, how would one interpret the VB \( | \psi_3 (1,2) \rangle \) ) and MO \( | \psi_5 (1,2) \rangle \) bonding wavefunctions? Which theory is more accurate for describing the dissociation of \(H_2\) (i.e., the molecule pulled apart) and why (look a your interpretations of the wavefunctions to address)?

     

     

     

    A proper wavefunction for a system of identical particles must reflects the requirement that the particles are indistinguishable from each other. This means interchanging the particles of any pair of states should not change the probability density (\( | \psi(1)(2) |^2\)) of the system. Hence

    \[ | \psi(1)(2) |^2 = | \psi(2)(1) |^2\]

    This divides all particles in nature into one of two classes with respect to exchange of electrons:

    • symmetric (bosons)

    \[ | \psi(1)(2) \rangle  = | \psi(2)(1) \rangle \]

    • antisymmetric (fermions):

    \[ | \psi(1)(2) \rangle  = -| \psi(2)(1) \rangle \]

    For each of the six wavefunctions above, identify if they are symmetric or anti-symmetric to exchange of the two electrons and if they accurately reflect \(H_2\).

     

     

     

     

     

    Although too long to be of use in the activity, you can get a lot of the answers in this video (https://www.youtube.com/watch?v=IcAdSETGUtg).