Orbital expansion

B3.1.5.2 THE LINEAR COMBINATIONS OF ATOMIC ORBITALS TO FORM MOLECULAR ORBITALS EXPANSION OF THE SPIN ORBITALS... 

For any sizeable system the Slater determinant can be tedious to write out, let alone the equivalent full orbital expansion, and so it is common to use a shorthand notation. Various notation systems have been devised. In one system the terms along the diagonal of the matrix are written as a single-row determinant. For the 3x3 determinant we therefore have ... 

If we substitute the atomic orbital expansion, we obtain a series of two-electron integrals, each of which involves four atomic orbitals ... 

We now introduce the atomic orbital expansion for the orbitals i/), and substitute for the corresponding spin orbital Xi into the Hartree-Fock equation,/,(l)x,(l) = X (1) ... 

When the Coulomb and exchange operators are expressed in terms of the basis functions and the orbital expansion is substituted for xu then their contributions to the Fock matrix element take the following form ... 

The CBS models use the known asymptotic convergence of pair natural orbital expansions to extrapolate from calculations using a finite basis set to the estimated complete basis set limit. See Appendix A for more details on this technique. 

The problem has now become how to solve for the set of molecular orbital expansion coefficients, c. . Hartree-Fock theory takes advantage of the variational principle, which says that for the ground state of any antisymmetric normalized function of the electronic coordinates, which we will denote H, then the expectation value for the energy corresponding to E will always be greater than the energy for the exact wave function ... 

The variational principle leads to the following equations describing the molecular orbital expansion coefficients, c. , derived by Roothaan and by Hall ... 

Both the Fock matrix—through the density matrix—and the orbitals depend on the molecular orbital expansion coefficients. Thus, Equation 31 is not linear and must be solved iteratively. The procedure which does so is called the Self-Consistent Field... 

So far, we have considered only the restricted Hartree-Fock method. For open shell systems, an unrestricted method, capable of treating unpaired electrons, is needed. For this case, the alpha and beta electrons are in different orbitals, resulting in two sets of molecular orbital expansion coefficients ... 

Realization with basis sets for Bloch orbital expansions that are physically, analytically and/or practically motivated, and also systematically improvable and testable ... 

Hartree-Fock (HF) calculations. This approximation imposes two constraints in order to solve the SchrOdinger equation and thus obtain the energy (a) a limited basis set in the orbital expansion and (b) a single assignment of electrons to orbitals. 

Owing to its complexity, the CC-R12 method was initially realized in various approximate forms. The first implementation of the CCSD-R12 method including noniterative connected triples [CCSD(T)-R12] was reported by Noga et al. [31,32,57-60] within the SA. The use of the same basis set for the orbital expansion and the RI in the SA rendered many diagrammatic terms to vanish and, thereby, drastically simplified the CCSD-R12 amplitude equations, easing its implementation effort. However, the simplified equations also meant that large basis sets (such as uncontracted quintuple- basis set) were needed to obtain reliable results and, therefore, the SA CCSD-R12 method was useful only in limited circumstances. 

All the above methods are somehow based on an orbital hypothesis. In fact, in the multipolar model, the core is typically frozen to the isolated atom orbital expansion, taken from Roothan Hartree Fock calculations (or similar [80]). Although the higher multipoles are not constrained to an orbital model, the radial functions are typically taken from best single C exponents used to describe the valence orbitals of a given atom [81]. Even tighter is the link to the orbital approach in XRCW, XAO, or VOM as described above. Obviously, an orbital assumption is not at all mandatory and other methods have been developed, for example those based on the Maximum Entropy Method (MEM) [82-86] where the constraints/ restraints come from statistical considerations. 

Harry Gray First, I want to comment on your point concerning the expansion of the d-orbitals to the outer sphere. I don t think there is any appreciable d-orbital expansion out this far, certainly not in the Co(CN)5H20r"2 case. 

Now, by its very definition, the global hardness rj is a measure of the HOMO-LUMO gap of a compound. Consequently, it seems reasonable to assume that the AEE AE should scale with the global hardness. The <(ag/r)3> term describes the p-orbital expansion or contraction as electrons are added or removed from the shielded nucleus. Using effective atomic numbers Znp and Slater rules ... 

The lower effective charge on the metal center causes an orbital expansion manifesting itself in a reduced interelectron repulsion. 

Fritsch, W. and Lin, C. D. (1984) Atomic-orbital-expansion studies of electron transfer in bare-nucleus Z (Z=2,4-8) -hydrogen-atom collisions, Phys. Rev. A, 29, pp. 3039-3051. 

Complete Basis Set Methods Petersson et al.61-63 developed a series of methods, referred to as complete basis set (CBS) methods, for the evaluation of accurate energies of molecular systems. The central idea in the CBS methods is an extrapolation procedure to determine the projected second-order (MP2) energy in the limit of a complete basis set. This extrapolation is performed pair by pair for all the valence electrons and is based on the asymptotic convergence properties of pair correlation energies for two-electron systems in a natural orbital expansion. As in G2 theory, the higher order correlation contributions are evaluated by a sequence of calculations with a variety of basis sets. 

Approximate solutions of the time-dependent Schrodinger equation can be obtained by using Frenkel variational principle within the PCM theoretical framework [17]. The restriction to a one-determinant wavefunction with orbital expansion over a finite atomic basis set leads to the following time-dependent Hartree-Fock or Kohn-Sham equation ... 

In contrast, the valence d and f orbitals in heavy atoms are expanded and destabilized by the relativistic effects. This is because the contraction of the s orbitals increases the shielding effect, which gives rise to a smaller effective nuclear charge for the d and f electrons. This is known as the indirect relativistic orbital expansion and destabilization. In addition, if a filled d or f subshell lies just inside a valence orbital, that orbital will experience a larger effective nuclear charge which will lead to orbital contraction and stabilization. This is because the d and f orbitals have been expanded and their shielding effect accordingly lowered. 

From molecular orbital theory, a many-electron wavefunction, may be defined by a determinant of molecular wavefunctions, i/ ,. The i[i, may in turn be expressed as a linear combination of one-electron functions, that is, (fj, = where cM, are the molecular orbital expansion coefficients... 

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