Matrix Hartree-Fock approximation

These relations show that the Fock-Dirac density matrix is identical with the first-order density matrix, and that consequently the first-order density matrix determines all higher-order density matrices and then also the entire physical situation. This theorem is characteristic for the Hartree-Fock approximation. 

The second-order density matrix is in the Hartree-Fock approximation given by Eqs. 11.44 and 11.53, and we obtain directly... 

Husimi, K., Proc. Phys.-Math. Soc. Japan 22, 264, "Some formal properties of the density matrix." Introduction of the concept of reduced density matrix. Statistical-mechanical treatment of the Hartree-Fock approximation at an arbitrary temperature and an alternative method of obtaining the reduced density matrices are discussed. 

In quantum chemistry, the correlation energy Ecorr is defined as Econ = exact HF- In Order to Calculate the correlation energy of our system, we show how to calculate the ground state using the Hartree-Fock approximation. The main idea is to expand the exact wavefunction in the form of a configuration interaction picture. The first term of this expansion corresponds to the Hartree-Fock wavefunction. As a first step we calculate the spin-traced one-particle density matrix [5] (IPDM) y ... 

See [11] for a recent review of applications of even-tempered basis set to the calculation of accurate molecular polarizabilities and hyperpolarizabilities within the matrix Hartree-Fock approximation. In [11] the results finite basis set Hartree-Fock calculations are compared with finite difference Hartree-Fock calculations. 

In other words, the Hartree-Fock approximation is nothing but setting the cumulant of the two-electron density matrix to be zero ... 

Within the Hartree-Fock approximation, calculations on molecules have almost all used the matrix SCF method, in which the HF orbitals are expanded in terms of a finite basis set of functions. Direct numerical solution of the HF equations, routine for atoms, has, however, been thought too difficult, but McCullough has shown that, for diatomic molecules, a partial numerical integration procedure will yield very good results.102 In particular, the Heg results agree well with the usual calculations, and it is claimed that the orbitals are likely to be of more nearly uniform accuracy than in the matrix HF calculations. Extensions to larger molecules should be very interesting so far, published results are available for He, Heg, and LiH. 

When using the Mpller-Plesset partitioning of the Hamiltonian, the zeroth order Hamiltonian Ho is defined by the Hartree-Fock approximation. For a nondegenerate ground state the matrix elements Vij of the one-particle part of the interaction in Eq. (44) are then given by... 

Analysis of Green s functions can be useful in seeking to establish model hamil-tonians with the property of giving approximately correct propagators, when put in the equations of motion. In this section, we explore a particularly simple model in order to familiarize the reader with various molecular orbital concepts using the terminology of Green s function theory. We employ the Hartree-Fock approximation and seek the molecular Fock operator matrix elements... 

The solution of Eq. (11.60) is achieved by standard methods once the matrix elements are obtained. Their evaluation involves the calculation of expectation values, which can be found when the solution of Eq. (11.60) is available, and we are, thus, faced with a self-consistency requirement, which is similar to but more complex than the corresponding challenge in the Hartree-Fock approximation. In the next section, the calculation of the matrix elements is addressed. 

There is a formal exact correspondence between the matrix elements fra and the Fock operator in the Hartree-Fock approximation, i.e., the expression... 

In calculations on molecules within the matrix Hartree-Fock approximation, it is found to be important to add polarization functions to double-zeta basis sets. Such basis functions do not improve the energies of the isolated component atomic species but contribute significantly to calculated bond energies and to the accuracy of calculated equilibrium bond angles. Double-zeta plus polarization basis sets (usually designated DZP or DZ + P) became widespread in quantum chemistry in the 1970s. In such a basis set the hydrogen atom is described by two s functions and one set of p functions the... 

Fig. 4. Potential energy curve for the ground state of the hydrogen molecule obtained by Kolos and Roo-thaan (Rev. Mod. Phys., 32, 169) (1960) using the Hartree-Fock approximation together with some energy values obtained by performing matrix Hartree-Fock calculations with a universal b set of elliptical functions.

Cooper and Wilson have investigated the calculation of spin-orbit coupling constants within the matrix Hartree-Fock approximation using systematic sequences of even-tempered basis sets of exponential-type functions. Some typical results are displayed in Table XVI. 

Function counterpoise corrections for the ground state of the neon atom calculated within the matrix Hartree-Fock approximation for a systematic sequence of even-tempered basis sets of Gaussian-type functions. In this table G represents a set of ghost orbitals. The internuclear separation in the NeG system is 5.0 bour."... 

This chapter introduces the basic concepts, techniques, and notations of quantum chemistry. We consider the structure of many-electron operators (e.g., the Hamiltonian) and discuss the form of many-electron wave functions (Slater determinants and linear combinations of these determinants). We describe the procedure for evaluating matrix elements of operators between Slater determinants. We introduce the basic ideas of the Hartree-Fock approximation. This allows us to develop the material of this chapter in a form most useful for subsequent chapters where the Hartree-Fock approximation and a variety of more sophisticated approaches, which use the Hartree-Fock method as a starting point, are considered in detail. 

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