Hartree-Fock, calculation

Even Hartree-Fock calculations are diflTicult and expensive to apply to large molecules. As a result, fiirther simplifications are often made. Parts of the Fock operator are ignored or replaced by parameters chosen by some sort of statistical procedure to account, in an average way, for the known properties of selected... 

This expression is not orbitally dependent. As such, a solution of the Hartree-Fock equation (equation (Al.3.18) is much easier to implement. Although Slater exchange was not rigorously justified for non-unifonn electron gases, it was quite successfiil in replicating the essential features of atomic and molecular systems as detennined by Hartree-Fock calculations. 

A simple example would be in a study of a diatomic molecule that in a Hartree-Fock calculation has a bonded cr orbital as the highest occupied MO (HOMO) and a a lowest unoccupied MO (LUMO). A CASSCF calculation would then use the two a electrons and set up four CSFs with single and double excitations from the HOMO into the a orbital. This allows the bond dissociation to be described correctly, with different amounts of the neutral atoms, ion pair, and bonded pair controlled by the Cl coefficients, with the optimal shapes of the orbitals also being found. For more complicated systems... 

Hartree-Fock Calculations for Atoms and Slater s Rules... 

A Hbasis functions provides K molecular orbitals, but lUJiW of these will not be occupied by smy electrons they are the virtual spin orbitals. If u c were to add an electron to one of these virtual orbitals then this should provide a means of calculating the electron affinity of the system. Electron affinities predicted by Konpman s theorem are always positive when Hartree-Fock calculations are used, because fhe irtucil orbitals always have a positive energy. However, it is observed experimentally that many neutral molecules will accept an electron to form a stable anion and so have negative electron affinities. This can be understood if one realises that electron correlation uDiild be expected to add to the error due to the frozen orbital approximation, rather ihan to counteract it as for ionisation potentials. 

The il/j in Equation (3.21) will include single, double, etc. excitations obtained by promoting electrons into the virtual orbitals obtained from a Hartree-Fock calculation. The second-order energy is given by ... 

The application of density functional theory to isolated, organic molecules is still in relative infancy compared with the use of Hartree-Fock methods. There continues to be a steady stream of publications designed to assess the performance of the various approaches to DFT. As we have discussed there is a plethora of ways in which density functional theory can be implemented with different functional forms for the basis set (Gaussians, Slater type orbitals, or numerical), different expressions for the exchange and correlation contributions within the local density approximation, different expressions for the gradient corrections and different ways to solve the Kohn-Sham equations to achieve self-consistency. This contrasts with the situation for Hartree-Fock calculations, wlrich mostly use one of a series of tried and tested Gaussian basis sets and where there is a substantial body of literature to help choose the most appropriate method for incorporating post-Hartree-Fock methods, should that be desired. 

Dor esi R, C Pisani, C Roetti and V R Saunders 1983. Treatment of Coulomb Interactions in Hartree-Fock Calculations of Periodic-Systems. Physical Review B28 5781-5792. 

Pisani C and R Dovesi 1980. Exact-Exchange Hartree-Fock Calculations for Periodic Systems. I. Illustration of the Method. International Journal of Quantum Chemistry XVII 501-516. 

A more complex set of functionals utilizes the electron density and its gradient. These are called gradient-corrected methods. There are also hybrid methods that combine functionals from other methods with pieces of a Hartree-Fock calculation, usually the exchange integrals. 

Another technique for obtaining an ionization potential is to use the negative of the HOMO energy from a Hartree-Fock calculation. This is called Koopman s theorem it estimates vertical transitions. This does not apply to methods other than HF but gives a good prediction of the ionization potential for many classes of compounds. 

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. 

There are also ways to perform relativistic calculations explicitly. Many of these methods are plagued by numerical inconsistencies, which make them applicable only to a select set of chemical systems. At the expense of time-consuming numerical integrations, it is possible to do four component calculations. These calculations take about 100 times as much CPU time as nonrelativistic Hartree-Fock calculations. Such calculations are fairly rare in the literature. 

Koopman s theorem a means for obtaining the ionization potential from a Hartree-Fock calculation... 

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). 

You will need to decide whether or not to request Restricted (RHF) or Unrestricted (UHF) Hartree-Fock calculations. This question embodies a certain amount of controversy and there is no simple answer. The answer often depends simply on which you prefer or what set of scientific prejudices you have. Ask yourself whether you prefer orbital energy diagrams with one or two electrons per orbital. 

Since the first formulation of the MO-LCAO finite basis approach to molecular Hartree-Fock calculations, computer applications of the method have conventionally been implemented as a two-step process. In the first of these steps a (large) number of integrals — mostly two-electron integrals — are calculated and stored on external storage. The second step then consists of the iterative solution of the Roothaan equations, where the integrals from the first step are read once for every iteration. 

DFT methods are attractive because they include the effects of electron correlation—the fact that electrons in a molecular system react to one another s motion and attempt to keep out of one another s way—in their model. Hartree-Fock calculations consider this effect only in an average sense—each electron sees and... 

Here we give the molecule specification in Cartesian coordinates. The route section specifies a single point energy calculation at the Hartree-Fock level, using the 6-31G(d) basis set. We ve specified a restricted Hartree-Fock calculation (via the R prepended to the HF procedure keyword) because this is a closed shell system. We ve also requested that information about the molecular orbitals be included in the output with Pop=Reg. 

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. 

As a final note, be aware that Hartree-Fock calculations performed with small basis sets are many times more prone to finding unstable SCF solutions than are larger calculations. Sometimes this is a result of spin contamination in other cases, the neglect of electron correlation is at the root. The same molecular system may or may not lead to an instability when it is modeled with a larger basis set or a more accurate method such as Density Functional Theory. Nevertheless, wavefunctions should still be checked for stability with the SCF=Stable option. ... 

We can compute all of the results except those in the first row by running just three jobs QCISD(T,E4T] calculations on HF and fluorine and a Hartree-Fock calculation on hydrogen (with only one electron, the electron correlation energy is zero). Note that the E4T option to the QCISDfT) keyword requests that the triples computation be included in the component MP4 calculation as well as in the QCISD calculation (they are not needed or computed by default). 

CBS models typically include a Hartree-Fock calculation with a very large basis set, an MP2 calculation with a medium-sized basis set (and this is also the level where the CBS extrapolation is performed), and one or more higher-level calculations with a medium-to-modest basis set. The following table outlines the components of the CBS-4 and CBS-Q model chemistries ... 

In general, DFT calculations proceed in the same way as Hartree-Fock calculations, with the addition of the evaluation of the extra term, This term cannot be evaluated analytically for DFT methods, so it is computed via numerical integration. 

Notice that NewZMat has set up a Hartree-Fock calculation by default, using the 6-31G(d) basis set. The molecule specification in the generated file is also in Z-matrix format rather than Cartesian coordinates. You can now edit this file to modify the procedure, basis set, and type of run desired. We won t bother running this file, since it is the same job as the one we just completed. 

Force Constants and Dipole-Moment Derivatives of Molecules from Perturbed Hartree-Fock Calculations I J. Gerratt and I. M. Mills Journal of Physical Chemistry 49 (1968) 1719... 

This is perhaps the easiest method to understand. It is based on the variational principle (Appendix B), analogous to the HF method. The trial wave function is written as a linear combination of determinants with the expansion coefficients determined by requiring that the energy should be a minimum (or at least stationary), a procedure known as Configuration Interaction (Cl). The MOs used for building the excited Slater determinants are taken from a Hartree-Fock calculation and held fixed. Subscripts S, D, T etc. indicate determinants which are singly, doubly, triply etc. excited relative to the... 

Hexahydropyiido[ 1,2-d -1,3,4-oxadiazines 212 may formally exist in three con-formers (Scheme 138) [99ACSA213]. However, according to simple Hartree-Fock calculations, the trans conformer 212c is the energetically preferred structure. 

Confusion is created by the often-quoted results of calculations by Latter that did predict some of the above ordering on the badis of the rather crude Thomas-Fermi method of approximation 20). More recent Hartree-Fock calculations on atoms show, for example, that the 3d level is definitely of lower energy than that of 4s (21). 

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