Integral-direct Hartree-Fock theory

An alternative to Hartree-Fock theory is density-functional (DF) theory in which certain elements in the Hamiltonian are evaluated at fixed points directly from the electron densities at those points. This can circumvent the need for calculating the enormous numbers of electron-electron repulsion integrals encountered in Hartree-Fock calculations. We here briefly describe density-functional theory based upon reviews by von Barth (1986) and Srivastava and Weaire (1987). 

The concept of purification is well known in the linear-scaling literature for one-particle theories like Hartree-Fock and density functional theory, where it denotes the iterative process by which an arbitrary one-particle density matrix is projected onto an idempotent 1-RDM [2,59-61]. An RDM is said to be pure A-representable if it arises from the integration of an Al-particle density matrix T T, where T (the preimage) is an Al-particle wavefiinction [3-5]. Any idempotent 1-RDM is N-representable with a unique Slater-determinant preimage. Within the linear-scaling literature the 1-RDM may be directly computed with unconstrained optimization, where iterative purification imposes the A-representabUity conditions [59-61]. Recently, we have shown that these methods for computing the 1 -RDM directly... 

Integration of the system of equations (9) yields trajectories of classical nuclei dressed with END. This approach can be characterized as being direct, and non-adiabatic or as fully non-linear time-dependent Hartree-Fock (TDHF) theory of quantum electrons and classical nuclei. This simultaneous dynamics of electrons and nuclei driven by their mutual instantaneous forces requires a different approach to the choice of basis sets than that commonly encountered in electronic structure calculations with fixed nuclei. This aspect will be further discussed in connection with applications of END. 

A particularly valuable addition to the arsenal of methods has been density functional theory (DFT), which aims to predict molecular properties with greater accuracy than Hartree-Fock calculations. Rather than directly integrating the electronic Schrodinger equation to get the AT-electron wavefunction (where N is the number of electrons in the molecule), DFT methods solve instead for the overall electron density, po. The many-electron wavefunction is a function of 3N coordinates Xi,yi,Zj for each electron i), but po depends on just 3 coordinates x, y, and z. Only this density function is needed to calculate the energy, and many other molecular properties. By skirting the need for the explicit, many-electron wavefunction, DFT methods provide a fast alternative route to predicting properties of molecules. Other methods make use of perturbation theory, variational configuration interaction, and extensions of valence bond theory. 

The next five chapters are each devoted to the study of one particular computational model of ab initio electronic-structure theory Chapter 10 is devoted to the Hartree-Fock model. Important topics discussed are the parametrization of the wave function, stationary conditions, the calculation of the electronic gradient, first- and second-order methods of optimization, the self-consistent field method, direct (integral-driven) techniques, canonical orbitals, Koopmans theorem, and size-extensivity. Also discussed is the direct optimization of the one-electron density, in which the construction of molecular orbitals is avoided, as required for calculations whose cost scales linearly with the size of the system. 

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