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Variational Minimization Viewpoint

The Variational Principle introduced and proven in Chapter 2 states that the trial wave function which yields the lowest energy expectation value is the one which is closest to the exact wave function. Therefore, it is logical to take a linear combination of Slater determinants or CSFs, functions which have been demonstrated to have many desirable properties as N-electron functions, and minimize the expectation value of the energy with respect to the linear expansion, a. k. a. CI, coefficients. Once again the minimization is taken subject to the constraint that the expansion functions remain orthonormal to one another, and the minimization method of choice is Lagrange's method of undetermined multipliers. In Lagrange's method, the maximum or minimum value of any functional f, in this case the energy expectation value, subject to the constraint, g, and dependent upon the set of variables [ $x_1 \cdots x_i \cdots x_n$] is found to occur when

\begin{displaymath}\delta L = \delta f + \lambda \delta g = 0, \end{displaymath} (14)

where in this case
f = $\displaystyle \langle \Psi \vert \hat{H} \vert \Psi \rangle$ (15)
g = $\displaystyle \langle \Psi \vert \Psi \rangle - 1 = 0$ (16)
L = $\displaystyle [ f + \lambda g] = \langle \Psi \vert \hat{H} \vert \Psi \rangle - \lambda \langle \Psi \vert \Psi \rangle - 1.$ (17)

Plugging the above definitions into the method of undetermined multipliers reveals that a minimum value of of the energy occurs when

\begin{displaymath}0 = \langle \delta \psi \vert \hat{H} \vert \psi \rangle + \l... ...vert \psi \rangle + \langle \psi \vert \delta \psi \rangle ]. \end{displaymath} (18)

Substituting into the linear expansion of determinant for $\Psi$ yields
$\displaystyle \langle \sum_{i}^{N} \delta C_i \phi_i \vert \hat{H} \vert \sum_{... ...{i}^{N} C_i \phi_i \vert \hat{H} \vert \sum_{j}^{N} \delta C_j \phi_j \rangle -$      
$\displaystyle E \left[ \langle \sum_{i}^{N} \delta C_i \phi_i \vert \sum_{j}^{N... ...sum_{i}^{N} C_i \phi_i \vert \sum_{j}^{N} \delta C_j \phi_j \rangle \right] = 0$     (19)

where E has been substituted in for $\lambda$ because, as in the SCF case, the undetermined multiplier is equal to the energy. Pulling the summations and the coefficients outside the square brackets gives
$\displaystyle \sum_{ij}^{N} \delta {C_i}^* C_j \langle \phi_i \vert \hat{H} \ve... ...{ij}^{N} {C_i}^* \delta C_j \langle \phi_i \vert \hat{H} \vert \phi_j \rangle -$      
$\displaystyle E \left[ \sum_{ij}^{N} \delta {C_i}^* C_j \langle \phi_i \vert \p... ...sum_{i}^{N} {C_i}^* \delta C_j \langle \phi_i \vert \phi_j \rangle \right] = 0.$     (20)

Collecting terms and utilizing the Hij and Sij notations first introduced in section 1.10 allow one to simplify the above equation down to two terms,

\begin{displaymath}\sum_{ij}^{N} \delta {C_i}^* C_j \left[ {H_i}_j - E {S_i}_j \... ...^{N} {C_i}^* \delta C_j \left[ {H_i}_j - E {S_i}_j \right] = 0 \end{displaymath} (21)

which are complex conjugates of one another. The addition of any pair of complex conjugates leaves only twice the real component that the pair have in common, and thus, with a little additional rearrangement the equation can be reduced to

\begin{displaymath}\sum_{i}^{N}{C_i}^* \left[ \sum_{j} {H_i}_j C_j - E{S_i}_j C_j \right] = 0. \end{displaymath} (22)

However, the $\delta {C_i}^*$ term is defined by Lagrange's method to be an arbitrary variation in Ci*; therefore, if the equation is to equal zero for all possible cases and not just the case where $\delta C_i =0$, the term contained within the bracket must be equal to zero:

\begin{displaymath}\sum_{j}^{N} {H_i}_j C_j - E {S_i}_j C_j = O \; \forall_i . \end{displaymath} (23)

This simplified minimization condition may be written as the matrix equation

\begin{displaymath}{\bf HC} = {\bf ESC}, \end{displaymath} (24)

which can be simplified one last time to

\begin{displaymath}{\bf HC} = {\bf EC} \end{displaymath} (25)

provided that the spin orbitals used to construct the Slater determinant are orthonormal. This final result found using the method of linear variations is equal to the prior result found using Heisenberg's matrix mechanics, equation (2.10), illustrating the equivalence of the two methods. Perhaps the best way to consolidate these two viewpoints is realize that only eigenvectors of the Hamiltonian matrix are stable with respect to variations of the linear expansion coefficients.
This page maintained by Brian C. Hoffman
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