Nonlocal orbitals

In all of the various VB methods that have been suggested involving nonlocal orbitals it is obvious that the orbitals must be written as linear combinations of AOs at many centers. Thus one is always faced with some sort of nonlinear minimization of the Rayleigh quotient. 

Historically, the first of the modem descendents of the Coulson-Fisher method proposed was the GGVB approach. Nevertheless, we will postpone its description, since it is a restricted version of still later proposals. 

Consider a system of n electrons in a spin state S. We know that there are for n linearly independent orbitals 

The SCVB energy is, of eourse, just the result from this optimization. Should a more elaborate wave funetion be needed, the virtual orbitals are available for a more-or-less eonventional, but nonorthogonal. Cl that may be used to improve the SCVB result. Thus an aeeurate result here may also involve a wave function with many terms. 

On the other hand, no such invariance of G1 or HLSP functions occurs, so the orthogonality constraint has a real impact on the calculated energy. 

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In 1949 Coulson and Fisher[15] introduced the idea of nonlocalized orbitals to the VB world. Since that time, suggested schemes have proliferated, all with some connection to the original VB idea. As these ideas developed, the importance of the spin degeneracy problem emerged, and VB methods frequently were described and implemented in this context. We discuss this more fully later. 

The reader will recall that in Chapter 2 we gave examples of H2 calculations in which the orbitals were restricted to one or the other of the atomic centers and in Chapter 3 the examples used orbitals that range over more than one nuclear center. The genealogies of these two general sorts of wave functions can be traced back to the original Heitler-London approach and the Coulson-Fisher[15] approach, respectively. For the purposes of discussion in this chapter we will say the former approach uses local orbitals and the latter, nonlocal orbitals. One of the principal differences between these approaches revolves around the occurrence of the so-called ionic structures in the local orbital approach. We will describe the two methods in some detail and then return to the question of ionic stmctures in Chapter 8. 

Huang, C. Carter, E. A. Nonlocal orbital-free kinetic energy density functional for semiconductors. Phys. Rev. B 2010,81,045206. 

The pseudopotential is derived from an all-electron SIC-LDA atomic potential. The relaxation correction takes into account the relaxation of the electronic system upon the excitation of an electron [44]- The authors speculate that ... the ability of the SIRC potential to produce considerably better band structures than DFT-LDA may reflect an extra nonlocality in the SIRC pseudopotential, related to the nonlocality or orbital dependence in the SIC all-electron potential. In addition, it may mimic some of the energy and the non-local space dependence of the self-energy operator occurring in the GW approximation of the electronic many body problem [45]. 

When is an eigenvalue of r(.B),. E is a pole. The corresponding operator, r(JS), is nonlocal and energy-dependent. In its exact limit, it incorporates all relaxation and differential correlation corrections to canonical orbital energies. A normalized DO is determined by an eigenvector of T Epou) according to... 

Several other molecular orbital models have been applied to the analysis of VCD spectra, primarily using CNDO wave functions. The nonlocalized molecular orbital model (NMO) is the MO analog of the charge flow models, based on atomic contributions to the dipole moment derivative (38). Currents are restricted to lie along bonds. An additional electronic term is introduced in the MO model that corresponds to s-p rehybridization effects during vibrational motion. 

Rotational and Dipole Strength Calculations for the CH-Stretching Vibrations of L-alanine Using the Localized Molecular Orbital, Nonlocalized Molecular Orbital, Atomic Polar Tensor, and Fixed Partial Charge Models ... 

Since H is specified, Eq. (5.3) defines Ec as a functional of the occupied orbitals of orbital functional derivative = v, (J), defines a nonlocal correlation potential vc in... 

The computational effort of solving orbital Euler-Lagrange (OEL) equations is significantly reduced if the generally nonlocal exchange-correlation potential vxc can be replaced or approximated by a local potential vxc(r). A variationally defined optimal local potential is determined using the optimized effective potential (OEP) method [380, 398]. This method can be applied to any theory in which the model... 

This argument shows that the locality hypothesis fails for more than two electrons because the assumed Frechet derivative must be generalized to a Gateaux derivative, equivalent in the context of OEL equations to a linear operator that acts on orbital wave functions. The conclusion is that the use by Kohn and Sham of Schrodinger s operator t is variationally correct, but no equivalent Thomas-Fermi theory exists for more than two electrons. Empirical evidence (atomic shell structure, chemical binding) supports the Kohn-Sham choice of the nonlocal kinetic energy operator, in comparison with Thomas-Fermi theory [288]. A further implication is that if an explicit approximate local density functional Exc is postulated, as in the local-density approximation (LDA) [205], the resulting Kohn-Sham theory is variation-ally correct. Typically, for Exc = f exc(p)p d3r, the density functional derivative is a Frechet derivative, the local potential function vxc = exc + p dexc/dp. 

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