Hartree field

In the work-interpretation derivation, the force field due to the pair-correlation density is first determined, and the potential then obtained as the work done to move an electron in this field. The force field and potential due to g (r,r p(r) are the Hartree field t nfr) and potential Wn(r) = Vnfr), respectively, since the spherically symmetric Fermi hole p r,F p(r) does not contribute to the field at the electron position. The field arises only due to the density p(r ) which is a charge distribution that is not spherically symmetric... 

Now the electronic density p(r) is a static charge distribution whose structure does not change as a function of electron position. Thus the Hartree field can be written as S h(t) = — VWH(r), where... 

To illustrate the calculation of x, we refer to work (see March, 1975) on the calculation of x(Z,Z) for neutral atoms from Hartree fields established for Z from 2 to 80. Foldy (see March, 1975) noted that if a /n — straight line results, which is well fitted by... 

Let us therefore suppose that we have set up an approximate potential field V for the molecule, in which an electron is to move. This field might, for example, be that obtained by the superposition of the Hartree fields of the component atoms. Let us also suppose that we are given a set of functions atomic orbitals of the various atoms of the molecule, although any set of independent functions could be used. The approximate orbitals can then be found by the method of trial eigenfunctions, using zero-order functions. The approximate energies of the molecular orbitals will therefore be the roots of the secular equation... 

Bobrowiez F W and Goddard W A III 1977 The self-eonsistent field equations for generalized valenee bond and open-shell Hartree-Foek wave funetions Modern Theoretical Chemistry vo 3, ed H F III Sehaefer (New York Plenum) pp 97-127... 

Field M J 1991 Constrained optimization of ab initio and semiempirical Hartree-Fock wavefunctions using... 

Direct dynamics attempts to break this bottleneck in the study of MD, retaining the accuracy of the full electronic PES without the need for an analytic fit of data. The first studies in this field used semiclassical methods with semiempirical [66,67] or simple Hartree-Fock [68] wave functions to heat the electrons. These first studies used what is called BO dynamics, evaluating the PES at each step from the elech onic wave function obtained by solution of the electronic structure problem. An alternative, the Ehrenfest dynamics method, is to propagate the electronic wave function at the same time as the nuclei. Although early direct dynamics studies using this method [69-71] restricted themselves to adiabatic problems, the method can incorporate non-adiabatic effects directly in the electionic wave function. 

Ah initio calculation s can be performetl at th e Ilartree-Fock level of approximation, equivalent to a self-con sisten t-field (SCK) calculation. or at a post llartree-Fock level which includes the effects of correlation —defined to be everything that the Hartree-Fock level of appi oxiniation leaves out of a n on-relativistic solution to the Schrddinger ec nation (within the clamped-nuclei Born-Oppenh e-imer approximation ). 

Drowicz F W and W A Goddard IB 1977. The Self-Consistent Field Equations for Generalized Valence Bond and Open-Shell Hartree-Fock Wave Functions. In Schaeffer H F III (Editor). Modem Theoretical Chemistry III, New York, Plenum, pp. 79-127. 

The Seetion on More Quantitive Aspects of Electronic Structure Calculations introduees many of the eomputational ehemistry methods that are used to quantitatively evaluate moleeular orbital and eonfiguration mixing amplitudes. The Hartree-Foek self-eonsistent field (SCF), eonfiguration interaetion (Cl), multieonfigurational SCF (MCSCF), many-body and Moller-Plesset perturbation theories. 

A configuration interaction calculation uses molecular orbitals that have been optimized typically with a Hartree-Fock (FIF) calculation. Generalized valence bond (GVB) and multi-configuration self-consistent field (MCSCF) calculations can also be used as a starting point for a configuration interaction calculation. 

Introductory descriptions of Hartree-Fock calculations [often using Rootaan s self-consistent field (SCF) method] focus on singlet systems for which all electron spins are paired. By assuming that the calculation is restricted to two electrons per occupied orbital, the computation can be done more efficiently. This is often referred to as a spin-restricted Hartree-Fock calculation or RHF. 

SCF (self-consistent field) procedure for solving the Hartree-Fock equations SCI-PCM (self-consistent isosurface-polarized continuum method) an ah initio solvation method... 

Tripos a molecular mechanics force field, also the name of a company that sells computational chemistry software TST (transition state theory) method for computing rate constants UHF (unrestricted Hartree-Fock)... 

In the RISM-SCF theory, the statistical solvent distribution around the solute is determined by the electronic structure of the solute, whereas the electronic strucmre of the solute is influenced by the surrounding solvent distribution. Therefore, the ab initio MO calculation and the RISM equation must be solved in a self-consistent manner. It is noted that SCF (self-consistent field) applies not only to the electronic structure calculation but to the whole system, e.g., a self-consistent treatment of electronic structure and solvent distribution. The MO part of the method can be readily extended to the more sophisticated levels beyond Hartree-Fock (HF), such as configuration interaction (Cl) and coupled cluster (CC). 

Ab initio calculations are iterative procedures based on self-consistent field (SCF) methods. Normally, calculations are approached by the Hartree-Fock closed-shell approximation, which treats a single electron at a time interacting with an aggregate of all the other electrons. Self-consistency is achieved by a procedure in which a set of orbitals is assumed, and the electron-electron repulsion is calculated this energy is then used to calculate a new set of orbitals, which in turn are used to calculate a new repulsive energy. The process is continued until convergence occurs and self-consistency is achieved." ... 

Hartree-Fock MO approach, the minimization of energy should provide the most accurate description of the electronic field. The mathematical problem is to define each of the terms, with being the most challenging. The formulation carmot be done exactly, but various approaches have been developed and calibrated by comparison with experimental data. The methods used most frequently by chemists were developed by A. D. Becke. " This approach is often called the B3LYP method. The computations can be done with... 

HF Hartree-Fock Self-Consistent Field Through 2nd derivatives... 

Gaussian also predicts dipole moments and higher multipole moments (through hexadecapole). The dipole moment is the first derivative of the energy with respect to an applied electric field. It is a measure of the asymmetry in the molecular charge distribution, and is given as a vector in three dimensions. For Hartree-Fock calculations, this is equivalent to the expectation value of X, Y, and Z, which are the quantities reported in the output. 

Gaussian can also predict some other properties dependent on the second and h er derivatives of the energy, such as the polarizabilities and hyperpolarizabilities. These depend on the second derivative with respect to an electric field, and are included automatically in every Hartree-Fock frequency calculation. 

I will refer to the Hartree model from time to time in the text. Hartree s energies were in poor agreement with experiment. With the benefit of hindsight he should have allowed for indistinguishability and the Pauli principle. This was Fock s contribution to the field he wrote the wavefunction as what we would now recognize as a Slater determinant. Such a wavefunction automatically satisfies the Pauli principle. 

---