Hamiltonian expectation values

Another potential problem is that the wave function may be contaminated not only by state s -F 1, but also states. v - - 2, s -F 3, etc. The s -F 1 annihilation operator will reduce the weights of these states in the annihilated wave function, but it will not eliminate them. Inspection of Eq. (C.33) should make clear that the higher states will contribute to the PUHF energy if they appear on both the left and right sides of the Hamiltonian expectation values with non-zero coefficients. When such contamination is important, recourse to a more complete projection operator, that annihilates an arbitrary number of spin states is available, but the computational cost increases to essentially that of an MP4 calculation. Note that the problems of the orbitals being non-ideal for the pure lowest spin state persist in this instance. 

The Hamiltonian expectation value over ) can be obtained in a similar... 

The dipole moment expectation value may be BCH expanded in the same way as the Hamiltonian expectation value ... 

If the transformation matrix is antisymmetric the Hamiltonian expectation value may be further rewritten as... 

The trial wavefunction depends on the elements of the antisymmetric matrix K, which determine the orbital variations, and on the CSF expansion coefficients c. The Hamiltonian expectation value for this wavefunction then takes the form... 

The Slaler-Condon rules tell us that the Hamiltonian expectation value for a single Slater determinant in which spin orbitals 0i,..., v are occupied is... 

A is a normalization constant), then this result can be used in Eq. (5.3) to express the Hamiltonian expectation value <0 // -f a/fJO) as a power series in a, upon which the desired second-order response is identified as the multiplier of (x. Of course, for each specific choice of the form of 0> (i.e., MCSCF, Cl, RSPT, CC) the prescription for evaluating Eq. (5.5) is different the basic approach is, however, identical for all such wavefunctions. 

The expression (11.75) is also valid in the Hartree-Fock approximation, where all the Vkr vanish. In the more general case, this expression may not agree with the direct calculation of the hamiltonian expectation value. This is a difficulty not uncommon in the application of Green s functions. It should have been avoided by some auxiliary constraint on the decoupling procedure . 

The simple form of Eq. 9.2 is due to the selection of as the PDF. The MC estimate of the denominator in the Hamiltonian expectation value is exactly one and has zero variance. Because is an approximate eigenfunction of the Hamiltonian,... 

A diagram may be used to clarify the situation (Figure 3.15). On the left, we place the orbital energies as calculated using a Hartree-Fock method. The substitution indicated by an arrow is placed along the horizontal scale according to the Hamiltonian expectation value h Oj The final energies after Cl are placed on... 

The total energy of the quantum-classical system is the smn of the quantum Hamiltonian expectation value with the classical kinetic and potential W(Q) energies... 

DMC with importance sampling yields the mixed distribution / = Pjo see Section 4.3. For operators that do not commute with the Hamiltonian, expectation values of o require the pure distribution 4>q- An approximate expression, valid to second order in the difference A =[Pg.1741]

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