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2.1: The Variational Method

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  • For the kind of potentials that arise in atomic and molecular structure, the Hamiltonian H is a Hermitian operator that is bounded from below (i.e., it has a lowest eigenvalue). Because it is Hermitian, it possesses a complete set of orthonormal eigenfunctions \(\{ |\psi_j \rangle\}\). Any function Φ that depends on the same spatial and spin variables on which H operates and obeys the same boundary conditions that the { \(\Psi\) j } obey can be expanded in this complete set

    \[Φ = \sum \limits_j C_j | \psi_j \rangle. \]

    The expectation value of the Hamiltonian for any such function can be expressed in terms of its \(C_j\) coefficients and the exact energy levels \(E_j\) of H as follows:

    \[\langle Φ| H |Φ\rangle = \sum\limits_{ij}C_iC_j \langle \psi_i |H| \psi_j \rangle = \sum\limits_j |C_j|^2 E_j .\]

    If the function Φ is normalized, the sum \(\sum\limits_j|C_j|^2\) is equal to unity. Because H is bounded from below, all of the \(E_j\) must be greater than or equal to the lowest energy \(E_0\) . Combining the latter two observations allows the energy expectation value of Φ to be used to produce a very important inequality:

    \[\langle Φ |H| Φ \rangle \geq E_0 .\]

    The equality can hold only if Φ is equal to \(\psi_0\) ; if Φ contains components along any of the other \(\psi_j\) , the energy of Φ will exceed \(E_0\).

    This upper-bound property forms the basis of the so-called variational method in which 'trial wavefunctions' Φ are constructed:

    1. To guarantee that Φ obeys all of the boundary conditions that the exact \(\Psi_j\) do and that Φ is of the proper spin and space symmetry and is a function of the same spatial and spin coordinates as the \(\Psi_j\);
    2. With parameters embedded in Φ whose 'optimal' values are to be determined by making \(\langle Φ |H| Φ \rangle\) a minimum.

    It is perfectly acceptable to vary any parameters in Φ to attain the lowest possible value for \(\langle Φ |H| Φ \rangle\) because the proof outlined above constrains this expectation value to be above the true lowest eigenstate's energy \(E_0\) for any Φ. The philosophy then is that the Φ that gives the lowest \(\langle Φ |H| Φ\rangle\) is the best because its expectation value is closes to the exact energy.

    Linear Variational Calculations

    Quite often a trial wavefunction is expanded as a linear combination of other functions (not the eigenvalues of the Hamiltonian, since they are not known)

    \[Φ = \sum_J^N C_J |Φ_J \rangle. \label{Ex1}\]

    In these cases, one says that a 'linear variational' calculation is being performed. The set of functions {\(Φ_J\)} are usually constructed to obey all of the boundary conditions that the exact state \(\Psi\) obeys, to be functions of the the same coordinates as \(Ψ\), and to be of the same spatial and spin symmetry as Ψ. Beyond these conditions, the {\(Φ_J\)} are nothing more than members of a set of functions that are convenient to deal with (e.g., convenient to evaluate Hamiltonian matrix elements \(\langle Φ_I|H|Φ_J \rangle\) that can, in principle, be made complete if more and more such functions are included in the expansion in Equation \(\ref{Ex1}\) (i.e., increase \(N\)).

    For such a trial wavefunction, the energy depends quadratically on the 'linear variational' \(C_J\) coefficients:

    \[\langle Φ |H| Φ \rangle = \sum_{I,J} ^{N,N}C_IC_J \langle Φ_Ι|H|Φ_J \rangle.\]

    Minimization of this energy with the constraint that Φ remain normalized, i.e.,

    \[\langle Φ|Φ \rangle = \sum\limits_{IJ} C_IC_J \langle Φ_I | Φ_J \rangle= 1\]

    gives rise to a so-called secular or eigenvalue-eigenvector problem:

    \[\sum\limits_J [\langle Φ_I|H|Φ_J \rangle - E \langle Φ_I|Φ_J \rangle] C_J = \sum\limits_J [H_{IJ} - E S_{IJ} ]C_J = 0. \]

    If the functions \(\{|Φ_J\rangle \}\) are orthonormal, then the overlap matrix \(S\) reduces to the unit matrix and the above generalized eigenvalue problem reduces to the more familiar form:

    \[ \sum\limits_J^N H_{IJ}C_J = E C_I .\]

    The secular problem, in either form, has as many eigenvalues \(E_i\) and eigenvectors {\(C_{iJ}\)} as the dimension of the \(H_{IJ}\) matrix as \(Φ\). It can also be shown that between successive pairs of the eigenvalues obtained by solving the secular problem at least one exact eigenvalue must occur (i.e.,\( E_{i+1} > E_{exact} > E_i\) , for all i). This observation is referred to as 'the bracketing theorem'.

    Variational methods, in particular the linear variational method, are the most widely used approximation techniques in quantum chemistry. To implement such a method one needs to know the Hamiltonian \(H\) whose energy levels are sought and one needs to construct a trial wavefunction in which some 'flexibility' exists (e.g., as in the linear variational method where the \(C_J\) coefficients can be varied). In Section 6 this tool will be used to develop several of the most commonly used and powerful molecular orbital methods in chemistry.

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