Sum-over-orbitals

Another quantity of some utility is the so-called local ionization potential, I(r). This is defined as the sum over orbital electron densities, pi(r) times absolute orbital energies, e i, and divided by the total electron density, p(r). 

In obtaining (5.16) from (5.15) we have used the fact that we can exchange the orbitals in both the bra and ket of a two-electron matrix element, which represents an integration over dummy coordinate—spin variables. Hhf is a one-electron operator since we sum over orbitals C). Equation (5.16) shows that we can diagonalise Hhf in the space of occupied orbitals only, since it has no matrix elements connecting occupied and unoccupied orbitals. Performing the diagonalisation we have... 

The case of helium gives a good test of theoretical methods, since there is only one target orbital in the Hartree—Fock approximation. Information is not further lost by a sum over orbitals. There have been several experiments on helium in different kinematic ranges. The distorted-wave Born approximation (McCarthy and Zhang, 1989) gives a good account of them. 

Ex involves a sum over orbital (a, a ) and spin (cr) quantum numbers, of a product of one-particle orbitals xjjaa with occupation numbers fa<, obeying Fermi statistics. 7[n] includes a spurious Coulomb self-interaction, as can be realized if Eq. (2) is applied to a single electron system. This Coulomb self-interaction is exactly cancelled out by the a = o terms in Ex (selfexchange term). 

The Uncoupled-Hartree-Fock method (UCHF) [31, 32, 50, 54, 85, 88, 100, 110] is also referred to as the sum-over-orbitals (SOO) method. In tliis technique, one takes the unperturbed Hamiltonian H° as a sum of one-particle Hamiltonians ... 

Hameka et al., has calculated polarizabilities up to third order for several organic systems [40, 76, 77, 78, 121, 122, 123, 124]. In his pioneering papers, the nonlinear polarizability was calculated by using sum over orbitals within the Hiickel approximation. This approach was later improved by using extended Hiickel method (EHM) [123, 124]. The quality of EHM third-order electric susceptibility was tested by... 

Bartkowiak, W., Strasburger, K., Leszczynski, J. Studie.s of molecular hyjierpolarizabilities (/3, y) for 4-nitroaniline (PNA). The application of quantum mechanical/Langevin dipoles/Monte Carlo (QM/LD/MC) and sum-over-orbitals (SOO) methods.. 1. Mol. Struct. (THELOCHEM) 549, 159-163 (2001)... 

Another (and more conventional) way of writing this is to represent it as a sum over orbitals, rather than electrons, as in equation (4-4). 

In fact, the equations of the SCF method may be cast into a form which emphasises this fundamental importance of the invariant R matrix. In this section it is shown that the orbital" form of the SCF equations is completely equivalent to an equation for the R matrix and vice versa. This form is particularly suitable for the application of perturbation theories which, typically, involve sums over orbitals not individual orbitals. 

Thus, magnetic field-like perturbations yield much easier response (or coupled perturbed ) equations in which flie contributions from any local potential vanish, hr fact, in the absence of HF exchange the A-matrix becomes diagonal and the linear equation system is trivially solved. This then leads to a sum-over-orbital -like equation for the second derivative that resembles in some way a sum-over-states equation. One should, however, carefully distinguish the sum-over-states picture from linear response or analytic derivative techniques since they have a very different origin. For electric field-like perturbations or magnetic field-like perturba-... 

The D+H2(t = l,i) reaction cross sections are shown in Fig. 5.7, as a function of total energy. The thick lines show the present calculations for j = (0,1,2,3), and the thin dotted line is the j = 0 result of Zhang and Miller obtained from the SMKVP [16]. We see complete agreement for j = 0 between the two methods over the entire energy range. The discrepancies between the SMKVP and the ABC-DVR-Newton method for J = 0 and E > 1 eV seem to have averaged out in the sum over orbital and total angular momentum. 

In Eq. (5.42), the factor of j - -1 counts the number of /-states in the sum over orbital angular momentum which contain a AT = 0 component. In Eq. (5.43), is the rotation constant of the linear transition state species, which is 8.6 cm for the LSTH PES description of D-I-H2. The above assumptions are expected to be satisfied at lower temperatures, but less so at higher temperatures. 

Thus E. is the average value of the kinetic energy plus the Coulombic attraction to the nuclei for an electron in ( ). plus the sum over all of the spin orbitals occupied in of the Coulomb minus exchange interactions. If is an occupied spin orbital, the temi [J.. - K..] disappears and the latter sum represents the Coulomb minus exchange interaction of ( ). with all of the 1 other occupied spin orbitals. If is a virtual spin orbital, this cancellation does not occur, and one obtains the Coulomb minus exchange interaction of cji. with all N of the occupied spin orbitals. 

The cross sum iif. which ts the sum over all the entries in a row and a column of atom i (= 2ii according to Eq. (1)) with the diagonal element h,- of atom i counted only once, indicates the total number of valence electrons in the orbitals of atom i (Eq. (4)). 

If you define a density matrix R by summing over all occupied molecular orbitals ... 

To obtaiti the total charge density r, at atom C we must sum over all occupied or partially occupied orbitals and subtract the I esult from 1,0. the n charge density of the carbon atom alone... 

Summing over the squares of the coefficients of the lower two orbitals (the upper orbital is unoccupied), we get electron densities of 1.502 at the terminal carbon atoms and 0.997 at the central atom. The charge densities on this iteration are... 

The sum over eoulomb and exehange interaetions in the Foek operator runs only over those spin-orbitals that are oeeupied in the trial F. Beeause a unitary transformation among the orbitals that appear in F leaves the determinant unehanged (this is a property of determinants- det (UA) = det (U) det (A) = 1 det (A), if U is a unitary matrix), it is possible to ehoose sueh a unitary transformation to make the 8i j matrix diagonal. Upon so doing, one is left with the so-ealled canonical Hartree-Fock equations ... 

The functions put into the determinant do not need to be individual GTO functions, called Gaussian primitives. They can be a weighted sum of basis functions on the same atom or different atoms. Sums of functions on the same atom are often used to make the calculation run faster, as discussed in Chapter 10. Sums of basis functions on different atoms are used to give the orbital a particular symmetry. For example, a water molecule with symmetry will have orbitals that transform as A, A2, B, B2, which are the irreducible representations of the C2t point group. The resulting orbitals that use functions from multiple atoms are called molecular orbitals. This is done to make the calculation run much faster. Any overlap integral over orbitals of different symmetry does not need to be computed because it is zero by symmetry. 

The coefficients are summed over the occupied orbitals only, and the factor of two comes from the fact that each orbital holds two electrons. 

Here, f/vai and Ucok are the Coulomb and exchange operators summed over the core and other occupied valence orbitals, respectively. 

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