Special Case The SCF Gradient Formula

The above gradient formula of Eq. (14.18) has been obtained by assuming that the wave function in question satisfies the formal Hellmann-Feynman theorem of Eq. (14.4) with respect to the second quantized Hamiltonian in the given basis set. Gradient formulae for diverse variational wave functions can be obtained as special cases of the general result of Eq. (14.18). In what follows we shall demonstrate how the familiar SCF gradient formula (Pulay 1969, Fletcher 1970, Pople et al. 1979) can be obtained from Eq. (14.18). The one-determinantal nature of the SCF wave function allows us to introduce some simplifications. Namely, as it was shown in Sect. 7.2  

Similarly, specifying the first- and second-order density matrices P and F in Eq. (14.18), one can obtain the actual gradient formulae for other types of variational wave functions as well. 

In this section we have presented a new derivation scheme for gradient formulae for all types of variational wave functions which violate the Hellmann-Feynman theorem only as a consequence of using an incomplete basis set. We have pointed out that if the second quantized Hamiltonian is applied, the Hellmann-Feynman theorem formally holds even in a finite basis, and the first derivatives of the energy can be obtained without considering wave function forces explicitly. On this ground, we derived a unified gradient formula from which the known results can be recovered in every special case. This has been demonstrated for the case of the Hartree-Fock wave function. 

The role of second quantization in this field is neither to contribute directly to practical gradient evaluations nor to give gradient formulae which cannot be derived by another means. The significance of this presentation is rather to offer an alternative and unifying view of energy derivatives, eliminating the concept of the wave function force for a rather wide class of variational functions used in quantum chemistry. The effects which lead to the wave function forces in the usual approach are included in the second quantized Hamiltonian. The two approaches are, therefore, completely equivalent and represent alternative possibilities to treat the problem. 

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