Atomic orbital integrals, calculation

In usual MO calculations with the ZORA Hamiltonian, the atomic orbital integrals derived from the ZORA Hamiltonian are simple and are evaluated numerically in direct space. In our study, however, we use the resolution of identity (RI) approximation with finite basis functions to evaluate them. To this end we use the relation. 

Direct The atomic orbital integrals are recomputed as needed. This does not require 0(N" ) internal or external storage but does involve additional computational effort. For large molecules, substantial savings are possible which compensate for this additional effort. However, direct methods are the only choice when memory and disk are exhausted and consequently are inevitably used for the largest calculations. 

SCF approximation. The indices //, v, A, and o denote four atomic orbital centers, so that the number of such orbitals that needs to be calculated increases proportionally scales with ) N, where N is the number of AOs, This was an intractable task in 1965, so Pople, Santry, and Segal introduced the approximation that only integrals in which = v and J. = o (i.e., li)) would be considered and that, further-... 

MP2 correlation energy calculations may increase the computational lime because a tw o-electron integral Iran sfonnalion from atomic orbitals (.40 s) to molecular orbitals (MO s) is ret]uired. HyperClicrn rnayalso need additional main memory arul/orcxtra disk space to store the two-eleetron integrals of the MO s. 

To this pom t, th e basic approxmi alien is th at th e total wave I lnic-tion IS a single Slater determinant and the resultant expression of the molecular orbitals is a linear combination of atomic orbital basis functions (MO-LCAO). In other words, an ah miiio calculation can be initiated once a basis for the LCAO is chosen. Mathematically, any set of functions can be a basis for an ah mitio calculation. However, there are two main things to be considered m the choice of the basis. First one desires to use the most efficient and accurate functions possible, so that the expansion (equation (49) on page 222). will require the few esl possible term s for an accurate representation of a molecular orbital. The second one is the speed of tW O-electron integral calculation. 

For both types of orbitals, the coordinates r, 0, and (j) refer to the position of the electron relative to a set of axes attached to the center on which the basis orbital is located. Although Slater-type orbitals (STOs) are preferred on fundamental grounds (e.g., as demonstrated in Appendices A and B, the hydrogen atom orbitals are of this form and the exact solution of the many-electron Schrodinger equation can be shown to be of this form (in each of its coordinates) near the nuclear centers), STOs are used primarily for atomic and linear-molecule calculations because the multi-center integrals < XaXbl g I XcXd > (each... 

Because the calculation of multi-center integrals that are inevitable for ab initio method is very difficult and time-consuming, Hyper-Chem uses Gaussian Type Orbital (GTO) for ab initio methods. In truly reflecting a atomic orbital, STO may be better than GTO, so HyperChem uses several GTOs to construct a STO. The number of GTOs depends on the basis sets. For example, in the minimum STO-3G basis set HyperChem uses three GTOs to construct a STO. 

The technique for this calculation involves two steps. The first step computes the Hamiltonian or energy matrix. The elements of this matrix are integrals involving the atomic orbitals and terms obtained from the Schrodinger equation. The most important con-... 

The second step determines the LCAO coefficients by standard methods for matrix diagonalization. In an Extended Hiickel calculation, this results in molecular orbital coefficients and orbital energies. Ab initio and NDO calculations repeat these two steps iteratively because, in addition to the integrals over atomic orbitals, the elements of the energy matrix depend upon the coefficients of the occupied orbitals. HyperChem ends the iterations when the coefficients or the computed energy no longer change the solution is then self-consistent. The method is known as Self-Consistent Field (SCF) calculation. 

The Extended Hiickel method neglects all electron-electron interactions. More accurate calculations are possible with HyperChem by using methods that neglect some, but not all, of the electron-electron interactions. These methods are called Neglect of Differential Overlap or NDO methods. In some parts of the calculation they neglect the effects of any overlap density between atomic orbitals. This reduces the number of electron-electron interaction integrals to calculate, which would otherwise be too time-consuming for all but the smallest molecules. 

All non-zero integrals over atomic orbitals on the two centers are set equal, as in CNDO/INDO, to an averaged y. Thus, (s sj s s ) = (s s I PbPb) = (PaPa I PbPb) = The two-center Coulomb integrals, rather than being calculated from first principles using s orbitals as in CNDO/INDO, are approximated by an Ohno-Klop-man [K.Ohno, Theor. Chim. Acta, 2, 219 (1964) G. Klopman,... 

In standard quantum-mechanical molecular structure calculations, we normally work with a set of nuclear-centred atomic orbitals Xi< Xi CTOs are a good choice for the if only because of the ease of integral evaluation. Procedures such as HF-LCAO then express the molecular electronic wavefunction in terms of these basis functions and at first sight the resulting HF-LCAO orbitals are delocalized over regions of molecules. It is often thought desirable to have a simple ab initio method that can correlate with chemical concepts such as bonds, lone pairs and inner shells. A theorem due to Fock (1930) enables one to transform the HF-LCAOs into localized orbitals that often have the desired spatial properties. 

The remaining integrals can be calculated from the functional form of the atomic orbitals. 

The CPHF equations are linear and can be determined by standard matrix operations. The size of the U matrix is the number of occupied orbitals times the number of virtual orbitals, which in general is quite large, and the CPHF equations are normally solved by iterative methods. Furthermore, as illustrated above, the CPHF equations may be formulated either in an atomic orbital or molecular orbital basis. Although the latter has computational advantages in certain cases, the former is more suitable for use in connection with direct methods (where the atomic integrals are calculated as required), as discussed in Section 3.8.5. 

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