Nonorthogonal spin orbitals

Our discussion so far has been concerned with the development of the second-quantization formalism for an orthonormal basis. Occasionally, however, we shall find it mote convenient to work with spin orbitals that are not orthogonal. We therefore extend the formalism of second quantization to deal with such spin orbitals, drawing heavily on the development in the preceding sections. 

Consider a set of nonorthogonal spin orbitals with overlap 

A Fock space for these spin orbitals can now be constructed as an abstract vector space using ON vectors as basis vectors in much the same way as for orthonormal spin orbitals. The inner product of the Fock space is defined such that, for vectors with the same number of electrons, it is equal to the overlap between the corresponding Slater determinants. For vectors with different particle numbers, the inner product is zero. The inner product is thus given by 

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McWeeny, R. and Sutcliffe, B. T., Methods of Molecular Quantum Mechanics 2nd ed., Academic Press, New York, 1976. Discusses the rules for evaluating matrix elements between Slater determinants formed from nonorthogonal spin orbitals. This book also contains a concise introduction to methods for obtaining spin eigenfunctions. 

The creation operators aj, for nonorthogonal spin orbitals are defined in the same way as for orthonormal spin orbitals (1.2.5). As for orthonormal spin orbitals, the anticommutation relations of the creation operators and the properties of their Hermitian adjoints (the annihilation operators) may be deduced from the definition of the creation operators and from the inner product (1.9.2). However, it is easier to proceed in the following manner. We introduce an auxiliary set of symmetrically orthonormalized spin orbitals... 

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. 

In this subsection, we will briefly discuss how one may construct a basis

carrier space which is adapted not only to the treatment of the ground state of the Hamiltonian H but also to the study of the lowest excited states. In molecular and solid-state theory, it is often natural and convenient to start out from a set of n linearly independent wave functions = < > which are built up from atomic functions (spin orbitals, geminals, etc.) involved and which are hence usually of a nonorthogonal nature due to the overlap of the atomic elements. From this set O, one may then construct an orthonormal set tp = d>A by means of successive, symmetric, or canonical orthonormalization.27 For instance, using the symmetric procedure, one obtains... 

This represents a formidable practical problem, as one is very unlikely to find isolated atoms with two nonorthogonal dipole moments and quantum states close in energy. Consider, for example, a V-type atom with the upper states 11), 3) and the ground state 2). The evaluation of the dipole matrix elements produces the following selection rules in terms of the angular momentum quantum numbers J — J2 = 1,0, J3 — J2 = 1,0, and Mi — M2 = M3 — M2 = 1,0. Since Mi / M3, in many atomic systems, p12 is perpendicular to p32 and the atomic transitions are independent. Xia et al. [62] have found transitions with parallel and antiparallel dipole moments in sodium molecules (dimers) and have demonstrated experimentally the effect of quantum interference on the fluorescence intensity. We discuss the experiment in more details in the next section. Here, we point out that the transitions with parallel and antiparallel dipole moments in the sodium dimers result from a mixing of the molecular states due to the spin-orbit coupling. 

There have been a number of means proposed for circumventing superposition error. Mayer et al. advocated what they term a chemical Hamiltonian approach, which separates the physical part of this operator from that responsible for BSSE using a nonorthogonal second quantization formalism. However, the physical Hamiltonian is no longer variational and the wavefunction is constructed from orthonormalized molecular spin orbitals. Surjan et al. " further developed this approach and performed pilot applications on small complexes. 

We normalize Pci such that the coefficient of is equal to 1. The spin orbitals (pa, natural spin orbitals of the respective lEPA pair correction function My 22) jhis will be the case for all of the methods considered and we shall not stress this again. One ought to add the abbreviation PNO (for Pair-Natural-Orbitals, sometimes interpreted as Pseudo-Natural-Orbitals) to the terms lEPA and CEPA as well, i.e. to speak of lEPA-PNO and CEPA-PNO rather than just of lEPA and CEPA. Using the PNO s one gets mutually orthogonal 0if though the PNO s of different pairs are nonorthogonal. Also the explicit expressions for Hab are not too complicated. 

The treatment of orbital overlap in conjunction with the use of nonorthogonal basis sets deserves particular attention in treatments in terms of electron field operators. The definition of creation and annihilation operators and their anticommutation rules are basic for this development. Let s( ) be a set of atomic spin orbitals used to define the creation operators... 

Since the nonorthogonal annihilation operators are linear combinations of orthonormal annihilation operators, we note that an annihilation operator times the vacuum state vanishes as for orthonormal spin orbitals ... 

It is noteworthy that the corresponding ex ession for a nonorthogonal creation operator working on an ON vector is identical to the expression for orthonormal spin orbitals (1.2.5)... 

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... 

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