Orbital nonorthogonal

More recently, the Landis group" has used valence bond concepts to derive new angular potential energy functions and to develop rules for parameterization. The VALBOND force field angular energy function is based on Pauling s hybrid orbital nonorthogonality functions and can be derived for any arbitrary combination of s, p, d, and f orbitals. 

Fattebert J-L and Bernholc J 2000 Towards grid-based 0(N) density-functional theory methods optimized nonorthogonal orbitals and multigrid acceleration Phys. Rev. B 62 1713-22... 

Pauling, L. Correlation of Nonorthogonality of Best Hybrid Bond Orbitals with Bond Strength of Orthogonal Orbitals Proc. Natl. Acad. Sci. (USA) 1976, 73, 274-275. 

The determinantal wave function in Eq. (21) is built [23] from complex dynamical spin orbitals Even when the basis orbitals ut in Eq. (22) are orthogonal these dynamical orbitals are nonorthogonal, and for a basis of nonorthogonal atomic orbitals based on Gaussians as those in Eq. (24) the metric of the basis becomes involved in all formulas and the END theory as implemented in the ENDyne code works directly in the atomic basis without invoking transformations to system orbitals. 

By virtue of the orthogonality of the transformation matrix T, the orbitals vk will then also form an orthonormal set. Nonorthogonal molecular orbitals... 

Such a spin-coupled wavefunction is optimized with respect to the core wavefunction (if applicable), as well as to the nonorthogonal valence bond orbitals,... 

We focus in this Section on particular aspects relating to the direct interpretation of valence bond wavefunctions. Important features of a description in terms of modem valence bond concepts include the orbital shapes (including their overlap integrals) and estimates of the relative importance of the different stmctures (and modes of spin coupling) in the VB wavefunction. We address here the particular question of defining nonorthogonal weights, as well as certain aspects of spin correlation analysis. 

As is well-known, modem valence-bond (VB) theory in its spin-coupled (SC) form (for a recent review, see Ref. 7) provides an alternative description of benzene [8-10] which, in qualitative terms, is no less convincing and is arguably even more intuitive than the MO picture with delocalized orbitals. The six n electrons are accommodated within a single product of six nonorthogonal orbitals, the spins of which are coupled in all five possible ways that lead to an overall six-electron singlet. The simultaneous optimization of the orbitals and of the weights of the five six-electron singlet spin... 

Throughout the Diels-Alder reaction, all six nonorthogonal SC orbitals resemble well-localized sp hybrids, each of which remains permanently attached to a single... 

G. K.-L. Chan and Troy Van Voorhis, Density-matrix renormahzation-group algorithms with nonorthogonal orbitals and non-Hermitian operators, and apphcations to polyenes. J. Chem. Phys. 122, 204101 (2005). 

The method appears stable and efficient and the choice of the most convenient values of the a and level shifter parameters did not require any particular care. The program has been tested by means of a code based on an iterative first-order Brillouin scheme where the nonorthogonality of the orbitals of the two Augments is directly taken into account by a general VB program, finding perfect agreement. 

The approach used first, historically, and the one this book is about, is called the valence bond (VB) method today. Heitler and London[8], in their treatment of the H2 molecule, used a trial wave function that was appropriate for two H atoms at long distances and proceeded to use it for all distances. The ideal here is called the separated atom limit . The results were qualitatively correct, but did not give a particularly accurate value for the dissociation energy of the H—H bond. After the initial work, others made adjustments and corrections that improved the accuracy. This is discussed folly in Chapter 2. A cmcial characteristic of the VB method is that the orbitals of different atoms must be considered as nonorthogonal. 

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. 

The above development shows that the effect of operators times an occupation number vector can be evaluated for the case, where the basis is nonorthogonal. The remaining problem in using nonorthogonal orbitals is how to efficiently calculate inner products between occupation number vectors. We will not describe that in any more detail in this book. 

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