Hartree-Fock calculations, current

Choose UHF (spin Unrestricted Hartree-Fock) or RHF (spin Restricted Hartree-Fock) calculations according to your molecular system. HyperChem supports UHF for both open-shell and closed-shell calculations and RHF for closed-shell calculations only. The closed-shell UHF calculation may be useful for studying dissociation of molecular systems. ROHF (spin Restricted Open-shell Hartree-Fock) is not supported in the current version of HyperChem (for ab initio calculations). 

For most molecules studied, modest Hartree-Fock calculations yield remarkably accurate barriers that allow confident prediction of the lowest energy conformer in the S0 and D0 states. The simplest level of theory that predicts barriers in good agreement with experiment is HF/6-31G for the closed-shell S0 state (Hartree-Fock theory) and UHF/6-31G for the open-shell D0 state (unrestricted Hartree-Fock theory). The 6-31G basis set has double-zeta quality, with split valence plus d-type polarization on heavy atoms. This is quite modest by current standards. Nevertheless, such calculations reproduce experimental barrier heights within 10%. 

Relativistic quantum chemistry is currently an active area of research (see, for example, the review volume edited by Wilson [102]), although most of the work is beyond the scope of this course. Much of the effort is based on Dirac s relativistic formulation of the Schrodinger equation this results in wave functions that have four components rather than the single component we conventionally think of. As a consequence the mathematical and computational complications are substantial. Nevertheless, it is very useful to have programs for Dirac-Fock (the relativistic analogue of Hartree-Fock) calculations available, as they can provide calibration comparisons for more approximate treatments. We have developed such a program and used it for this purpose [103]. 

Hartree-Fock calculations are sometimes said to ignore, or at least to neglect, electron correlation. Actually, the Hartree-Fock method allows for some electron correlation according to our current understanding, two electrons of the same spin... 

Ground State of CO. The CO molecule has been extensively studied, both experimentally and theoretically. Table 14 compares ground-state 02+) spectroscopic constants calculated by the Hartree-Fock method109 and by the density functional approach99 with experiment.104 In addition to these spectroscopic constants, the polar nature of the molecule provides a further measurable quantity, the dipole moment. Since the intensities of infrared vibration-rotation bands allow the dipole moment to be determined as a function of C-O separation this provides a useful comparison with the results of ab initio calculations. For example, the positive sign obtained from the equilibrium dipole moment by Hartree-Fock calculations was viewed as a reason to question the negative value found experimentally, whereas the current view is that the positive sign is a defect of the Hartree-Fock method. 

In principle, using the above formalism the current density and the transfer matrix element can be evaluated at any level of the electronic structure calculations. Here we consider the Hartree-Fock calculations. 

Functional forms based on the above ideas are used in the HFD [127] and Tang-Toermies models [129], where the repulsion term is obtained by fitting to Hartree-Fock calculations, and in the XC model [92] where the repulsion is modelled by an ab initio Coulomb term f and a semi-empirical exchange-repulsion term Current versions of all these models employ an individually damped dispersion series for the attractive... 

The previous section outlined the development of correlation consistent basis sets involving mostly light, p-block elements. In the extension of these ideas to heavier elements, the effects of relafivify on fhe basis set should be introduced. In addition to relativistic effects, the influence of low-lying electronic states must also be considered in the cases of fhe fransifion metals. For most cases only scalar relativistic effects will be considered, i.e., even if spin-orbit coupling is included in the calculations, each i component will be described by the same contracted basis set. The exceptions are the correlation consistent basis sets of Dyall [29-33], which were developed in fully-relafivisfic, 4-component Dirac-Hartree-Fock calculations. These basis sets, which are of DZ-QZ quality, are currently available for the heavier p-block elements, as well as the 4d and 5d transition metals. 

This is not the place for a full overview of the wave function based post Hartree-Fock methods currently applied for the calculation of intermolecular interactions and in particular molecule/surface interactions. Table 3 contains a brief characterization of the most widely applied schemes. The two most popular methods are MP2 (second order Moller-Plesset perturbation theory), because it covers large part of electronic correlation at comparably low ex-... 

The relativistic form of the one-electron Schrodinger equation is the Dirac equation. One can do relativistic Hartree-Fock calculations using the Dirac equation to modify the Fock operator, giving a type of calculation called Dirac-Fock (or Dirac-Hartree-Fock). Likewise, one can use a relativistic form of the Kohn-Sham equations (15.123) to do relativistic density-functional calculations. (Relativistic Xa calculations are called Dirac-Slater or Dirac-Xa calculations.) Because of the complicated structure of the relativistic KS equations, relatively few all-electron fully relativistic KS molecular calculations that go beyond the Dirac-Slater approach have been done. [For relativistic DFT, see E. Engel and R. M. Dreizler, Topics in Current Chemistry, 181,1 (1996).]... 

Having described restricted closed-shell Hartree-Fock calculations in conjunction with the H2 and HeH model systems, we now want to illustrate the results of more realistic calculations on polyatomic molecules. We do this not to provide a review of current calculations, but rather to illustrate the main ideas behind all calculations of the closed-shell restricted Hartree-Fock type, and to provide some feeling and intuition for how well (or how... 

Quantum chemistry for lanthanides and actinides is an active area of current research. The applicable methods range from relativistically parametrized semiempirical extended Hiickel-type approaches to fully relativistic allelectron Dirac-Hartree-Fock calculations with a subsequent correlation treatment. It is emphasized that electron correlation effects and relativistic effects including spin-orbit coupling have to be treated simultaneously in order to avoid errors arising from the nonadditivity of these effects. Considerable progress is expected, especially on the ab initio side of quantum chemical applications, for small lanthanide and actinide systems during the next few years. 

The problem with most quantum mechanical methods is that they scale badly. This means that, for instance, a calculation for twice as large a molecule does not require twice as much computer time and resources (this would be linear scaling), but rather 2" times as much, where n varies between about 3 for DFT calculations to 4 for Hartree-Fock and very large numbers for ab-initio techniques with explicit treatment of electron correlation. Thus, the size of the molecules that we can treat with conventional methods is limited. Linear scaling methods have been developed for ab-initio, DFT and semi-empirical methods, but only the latter are currently able to treat complete enzymes. There are two different approaches available. 

The final application considered in this chapter is chosen to illustrate the application of a QM-MM study of an enzyme reaction that employs an ab initio Hamiltonian in the quantum region [67]. Because of the computational intensity of such calculations there are currently very few examples in the literahire of QM-MM shidies that use a quanhim mechanical technique that is more sopliisticated than a semiempirical method. MuUiolland et al. [67] recently reported a study of part of the reaction catalyzed by citrate synthase (CS) in wliich the quanhim region is treated by Hartree-Fock and MP2 methods [10,51],... 

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