Hamiltonian 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. 

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-... 

Tike all effective one-electron approaches, the mean-field approximation considerably quickens the calculation of spin-orbit coupling matrix elements. Nevertheless, the fact that the construction of the molecular mean-field necessitates the evaluation of two-electron spin-orbit integrals in the complete AO basis represents a serious bottleneck in large applications. An enormous speedup can be achieved if a further approximation is introduced and the molecular mean field is replaced by a sum of atomic mean fields. In this case, only two-electron integrals for basis functions located at the same center have to be evaluated. This idea is based on two observations first, the spin-orbit Hamiltonian exhibits a 1/r3 radial dependence and falls off much faster... 

The one-center approximation allows for an extremely rapid evaluation of spin-orbit mean-field integrals if the atomic symmetry is fully exploited.64 Even more efficiency may be gained, if also the spin-independent core-valence interactions are replaced by atom-centered effective core potentials (ECPs). In this case, the inner shells do not even emerge in the molecular orbital optimization step, and the size of the atomic orbital basis set can be kept small. A prerequisite for the use of the all-electron atomic mean-field Hamiltonian in ECP calculations is to find a prescription for setting up a correspondence between the valence orbitals of the all-electron and ECP treatments.65-67... 

It is often assumed that resonance integrals are proportional to the overlap between the atomic orbitals which formally underlie the pi Hamiltonian [58]. If these are assumed to be ordinary 2p atomic orbitals for carbon, the distance dependence of the overlap can be calculated analytically, and the distance dependence of the t parameters is often taken to be of this form, as for example in Extended Hiickel Theory [59]. But there is no need to make this assumption since the parameters in the pi Hamiltonian should more properly be thought of as rescaled effective integrals, and there is evidence that the model performs better if the t values are allowed to vary more rapidly with distance. Accordingly, we have adopted the form... 

The DV-Xa molecular orbital calculational method used here utilizes basis sets of numerically calculated atomic orbitals, as well as those of analytical atomic orbitals such as Slater orbitals. Matrix element of the Hamiltonian and the overlap integral are calculated numerically by summing integrand at sampling points rk, the Diophantine points, which are distributed according to the weighted function, and expressed as. 

The integrals are calculated in terms of the atomic orbitals (AOs) and are subsequently transformed to the orthonormal basis. In some cases it may be more efficient to evaluate the expressions in the nonorthogonal AO basis. We return to this problem when we consider the calculation of the individual geometry derivatives. For the time being we assume that the Hamiltonian is expressed in the orthonormal molecular orbital (MO) basis. The second-quantized Hamiltonian [Eq. (8)] is a projection of the full Hamiltonian onto the space spanned by the molecular orbitals p, i.e., the space in which calculations are carried out. 

In electronic structure calculations, it is not unlikely for a basis set to be dependent on the parameters. The most obvious case involves geometric parameters. The atomic orbital basis functions used to construct molecular orbitals are generally chosen to follow the atomic centers. This means that the functions are dependent on the molecular geometry, and so there will be nonzero derivatives of the usual one- and two-electron integrals. In the case of parameters such as an electric field strength, there is no functional dependence of the standard types of basis functions. The derivatives of all the basis functions with respect to this parameter are zero, and so all derivative integrals involving the zero-order Hamiltonian terms are zero as well. 

As discussed in section 2.3, the Hamiltonian matrix elements are generally written in terms of one- and two-electron integrals in the molecular orbital (MO) basis. However, these integrals are originally calculated in the atomic orbital (AO) basis, or perhaps the symmetry-adapted orbital (SO) basis. Therefore it is necessary to transform the AO or SO integrals into the MO basis, according to... 

H was the matrix-component of the Hiickel effective-Hamiltonian operator, effective between two basis atomic-orbitals, 4>r and 4>s, Srs was the overlap integral between 4>r and s, and H was set equal to a, H to / . This is how we developed the simple HMO-approach in Chapter Two. What Roothaan did was to show that a formally similar determinant is obtained in a full treatment of the re-electrons, but that it involves a somewhat more complicated expression for the matrix-elements, H . Furthermore, he showed that this more-complicated expression somehow had to take into account interactions between any one re-electron and all the other re-electrons. We do not go into the details of this here, except to say that, in order to find the LCAO-MO coefficients for one molecular orbital, it is necessary to know all the others, because all the others appear in the expressions for the equivalent terms, Hrs. This is a very familiar situation which mathematicians have long known how to deal with and which we encountered during our discussion of the self-consistent" Huckel-methods in 7.2—7.5 it is necessary to use an iterative scheme. An initial guess is made of all the orbitals except one and these are used to calculate the H -terms for the one orbital which has not yet... 

PPP (Pariser-Parr-Pople) [14-16] is an SCF (self-consistent field) Jt-electron theory, assuming o — jt separability. Only a single (2pz) atom orbital is considered on each atom and the Ji-electron Hamiltonian includes electron-electron interactions with ZDO (zero differential overlap) approximation. All integrals are determined by semiempirical parameters. The PPP method can only be used to calculate those physical properties for which jt electrons are mainly responsible. 

The use of spherically symmetric s orbitals avoids the problems associated with transformations of the axes. The core Hamiltonians (Lf ) are not calculated but are obtained from experimental ionisation energies. This is because it is important to distinguish between s and p orbitals in the valence shell (i.e. the 2s and 2p orbitals for the first-row elements), and without explicit core electrons this is difficult to achieve. The resonance integrals, /3ab/ 3re written in terms of empirical single-atom values as follows ... 

We have already mentioned that the Cl method converges slowly. Due to this, the Hamiltonian matrices and overlap integral matrices are sometimes so large that they cannot fit into the computer memory. In practice, such a situation occurs in all good quality calculations for small systems and in all calculations for medium and large systems. Even for quite large atomic orbital basis, the number of integrals is much smaher than the number of Slater determinants in the Cl expansion. 

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