Hartree-Fock quantum-mechanical method

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],... 

MOPAC is a general-purpose semiempirical molecular orbital program for the study of chemical structures and reactions. It is available in desktop PC running Windows, Macintosh OS, and Unix-based workstation versions. It uses semiempirical quantum mechanical methods that are based on Hartree-Fock (HF) theory with some parameterized functions and empirically determined parameters replacing some sections of the complete HF treatment. The approximations in... 

During the last decade MO-theory became by far the most well developed quantum mechanical method for numerical calculations on molecules. Small molecules, mainly diatomics, or highly symmetric structures were treated most accurately. Now applicability and limitations of the independent particle, or Hartree-Fock (H. F.), approximation in calculations of molecular properties are well understood. An impressive number of molecular calculations including electron correlation is available today. Around the equilibrium geometries of molecules, electron-pair theories were found to be the most economical for actual calculations of correlation effects ). Unfortunately, accurate calculations as mentioned above are beyond the present computational possibilities for larger molecular structures. Therefore approximations have to be introduced in the investigation of problems of chemical interest. Consequently the reliability of calculated results has to be checked carefully for every kind of application. Three types of approximations are of interest in connection with this article. 

Therefore the scaling transformation of the quantum-mechanical force field is an empirical way to account for the electronic correlation effects. As far as the conditions listed above are not always satisfied (e.g. in the presence of delocalized 7r-electron wavefunctions) the real transformation is not exactly homogeneous but rather of Puley s type, involving n different scale constants. The need of inhomogeneous Puley s scaling also arises due to the fact that the quantum-mechanical calculations are never performed in the perfect Hartree-Fock level. The realistic calculations employ incomplete basis sets and often are based on different calculation schemes, e.g. semiempirical hamiltonians or methods which account for the electronic correlations like Cl and density-functional techniques. In this context we want to stress that the set of scale factors for the molecule under consideration is specific for a given set of internal coordinates and a given quantum-mechanical method. 

C. Density functional theory Density functional theory (DFT) is the third alternative quantum mechanics method for obtaining chemical structures and their associated energies.Unlike the other two approaches, however, DFT avoids working with the many-electron wavefunction. DFT focuses on the direct use of electron densities P(r), which are included in the fundamental mathematical formulations, the Kohn-Sham equations, which define the basis for this method. Unlike Hartree-Fock methods of ab initio theory, DFT explicitly takes electron correlation into account. This means that DFT should give results comparable to the standard ab initio correlation models, such as second order M(j)ller-Plesset (MP2) theory. 

Fig. 1. Schematic representation of the potential energy surface for the electronic (el) ground state of a molecule existing in two tautomeric forms, A and B. Superscripts exp, HF, CNDO/2, MINDO/3 indicate that energy differences 8 a,b calculated for potential energy surfaces determined either experimentally (exp) or calculated by means of ab initio method in the Hartree-Fock (HF) approximation or by semiempirical methods (CNDO/2, MINDO/3). The symbol eq stands for the geometrical equilibrium of both tautomers, while 2a and Qb indicate nonequilibrium geometries of tautomers A and B, respectively. Note that the theoretical potential surface calculated by sophisticated quantum-mechanical methods ( exact solution of electronic Schrbdinger equation includes electron correlation with geometry optimization) should be the same (or very similar) as that determined experimentally [in this case i>eor) ei

Two general quantum mechanical procedures were applied to the calculation of atomic electron affinities prior to 1960. These were the variational method and Hartree Fock semi-empirical method. Prior to 1960 there were no accurate calculations for elements other than hydrogen. Indeed, only the electron affinities of C and... 

In this review, I have interpreted the term Car-Parrinello methods in the broad sense to mean those which combine first-principles quantum mechanical methods with molecular dynamics methods. I use this term synonymously with uh initio molecular dynamics, first-principles molecular dynamics, and ab initio simulations. Thus, ways of solving the many-body electronic problem, such as Hartree-Fock and correlation methods, are included, in addition to the projector-augmented plane-wave method. In the original Car-Parrinello method, molecular motion is treated classically via... 

Computational quantum mechanical methods, such as the Hartree-Fock method (Hehre et al., 1986 Szabo and Ostlund, 1989 Levine, 2000), were developed to convert the many-body Schrodinger equation into a singleelectron equation, which can then be solved tractably with modern computational power. The single-electron equation is an approach by which the state (or wavefunction) of each electron is computed within the field... 

This description of quantum mechanical methods for computing (hyper)polarizabilities demonstrates why, nowada, the determination of hyperpolarizabilities of systems containing hundreds of atoms can, at best, be achieved by adopting, for obvious computational reasons, semi-empirical schemes. In this study, the evaluation of the static and dynamic polarizabilities and first hyperpolaiizabilities was carried out at die Time-Dependent Hartree-Fock (TDOT) [39] level with the AMI [50] Hamiltonian. The dipole moments were also evaluated using the AMI scheme. The reliability of the semi-empirical AMI calculations was addressed in two ways. For small and medium-size push-pull polyenes, the TDHF/AMl approach was compared to Hartree-Fock and post Hartree-Fock [51] calculations of die static and dynamic longitudinal first hyperpolarizability. Except near resonance, the TDHF/AMl scheme was shown to perform appreciably better than the ab initio TDHF scheme. Then, the static electronic first hyperpolaiizabilities of the MNA molecule and dimer have been calculated [15] with various ab initio schemes and compared to the AMI results. In particular, the inclusion of electron correlation at the MP2 level leads to an increase of Paaa by about 50% with respect to the CPHF approach, similar to the effect calculated by Sim et al. [52] for the longitudinal p tensor component of p-nitroaniline. The use of AMI Hamiltonian predicts a p aa value that is smaller than the correlated MP2/6-31G result but larger than any of the CPHF ones, which results fi-om the implicit treatment of correlation effects, characteristic of die semi-empirical methods. This comparison confirms that a part of die electron... 

Semiempirical molecular quantum-mechanical methods use a simpler Hamiltonian than the correct molecular Hamiltonian and use parameters whose values are adjusted to fit experimental data or the results of ab initio calculations an example is the Hiickel MO treatment of conjugated hydrocarbons (Section 16.3), which uses a one-electron Hamiltonian and takes the bond integrals as adjustable parameters rather than quantities to be calculated theoretically. In contrast, an ab initio (or first principles) calculation uses the correct Hamiltonian and does not use experimental data other than the values of the fundamental physical constants. A Hartree-Fock SCF calculation seeks the antisymmetrized product d> of one-electron functions that minimizes / dr, where H is the true Hamiltonian, and is thus an ab initio calcula-... 

The solvent-induced stereochemical behaviour of a bile acid-based biphenyl phosphite has been studied experimentally using circular dichroism (CD) spectroscopy, and theoretically using DFT quantum mechanical methods. " The FTIR, Raman and surface-enhanced Raman scattering (SERS) spectra of phenyl phosphate disodium salt have been recorded and its vibrational wavenumbers, calculated using the Hartree-Fock/6-31G basis set, compared with experimental values. From SERS spectra study, the molecule is adsorbed on the silver surface with the benzene ring in a tilted orientation. The presence of the phenyl ring and the phosphate group vibrations in the SERS spectrum reveal the interactions between the phenyl... 

The van der Waals parameters have usually been obtained by fitting to crystal structure data and to heats of sublimation of molecular crystals,77,93-97,101-103 but it is also possible to obtain parameters from quantum mechanical calculations. Because van der Waals interactions include dispersion, it would be necessary to use a correlated quantum mechanical method. Unfortunately, density functional methods do not describe dispersive interactions, and post-Hartree-Fock methods, capable of capturing this physical effect, are computationally expensive. The dispersion interactions are therefore often ignored. The repulsive part of the van der Waals interaction, on the other hand, can be determined from test particle calculations - using the Hartree-Fock method. 

Quantum mechanical calculations view a molecule as a collection of point nuclei and electrons with fixed masses and charges. The energy terms include the kinetic energy of each particle and the coulombic energies between the particles (repulsion between nuclei, attraction between the nucleus and an electron, and repulsion between electrons). Here we will review some basic equations of quantum mechanics to understand quantum mechanical methods, such as Hartree-Fock, semiempirical, and density functional theory methods, that have been most widely used for the clay mineral modeling where simulation size is greater than the molecular cluster with several atoms. 

Much like the choice ot the empirical potential functional form discussed in Sect. 2 above, the choice of the quantum mechanical method and model is a compromise between speed of evaluation and accuracy. The most rigorous approach to evaluating these energy and force terms would be to use ab initio quantum chemical methods with large basis sets and correlation corrections beyond the Hartree Fock level. Clearly, this is currently not a feasible approach because of the computional demands such as model places on a single energy evaluation, not to mention the iterative evaluation over thousands of structures (timesteps) of a dynamics simulation. 

The Hartree-Fock MO theory is the most extensively tested and documented quantum mechanical method. The method can easily be extended to... 

It is clear that the inclusion of correlation at appropriate levels accounts for virtually all the disagreement between theory and experiment found at the Hartree-Fock level in correlation-sensitive molecules. It remains to be seen how efficiently some of the advanced post-Hartree-Fock methods can be implemented to handle larger molecules. One of the major advantages of density functional theory is its speed relative to conventional quantum mechanical methods. If it can be extended to give somewhat better agreement with experiment, it may well be the method of choice for treating large chemical systems in the near future. 

There is an important lesson for computational chemistry in this derivation itself we have shown that in order to calculate the effects of dispersion forces quantitatively, it is not enough to know the electron distribution in the ground state. For this reason, Hartree-Fock calculations, which in many respects are powerful predictors of molecular properties, cannot calculate dispersion forces. This limitation severely restricts our ability to predict properties of large molecules and large groups of molecules by quantum-mechanical methods. 

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