Orthogonalization 4-spinors

In this systematic construction process, spinors which enter the reference Slater determinant (these are the so-called occupied spinors) are substituted by new orthogonal spinors (which are, for instance, the virtual spinors that are obtained in a Dirac-Fock-Roothaan calculation see chapter 10). One then usually distinguishes sets of singly substituted many-electron functions 4>o, where an occupied spinor o, has been substituted by a virtual spinor from doubly substituted o]o j so on. The o are also called single-excited determinants, the I oj-oy then double-excited determinants etc. It is important to understand that these excitations are not to be confused with the excited states of quantum mechanics they simply denote a substitution pattern. Eq. (8.100) can be rewritten in terms of the excitation hierarchy as... 

It should be noted that these two solutions are orthogonal to each other uftPWP) = 0. By noting that the projection operator P+(p) operating on the remaining two basis spinors... 

The molecular four-component spinors constructed this way are orthogonal to the inner core spinors of the atom, because the atomic basis functions used in Eq. (6.4) are generated with the inner core shells treated as frozen. 

Apart from the phase factor — i, the transformed spinor will be recognized as the ungerade spinor of Chapter 11. The original and time-reversed states are orthogonal and therefore degenerate, and consequently... 

The operators A constructed in the course of this procedure form an orthogonal set, Aj-At oc 6ij. In Eqs. (3) and (4) the definition of the inner product Ai At depends on the type of the considered Green s function. For example, for functions (2) these are ( A, At ) and ([A, At]), respectively. The method described by Eqs. (3) and (4) can be straightforwardly generalized to the case of many-component operators which is necessary, for example, to consider Green s functions for Nambu spinors in the superconducting state [7]. 

Representation theory of molecular point groups tells us how a rotation or a reflection of a molecule can be represented as an orthogonal transformation in 3D coordinate space. We can therefore easily determine the irreducible representation for the spatial part of the wave function. By contrast, a spin eigenfunction is not a function of the spatial coordinates. If we want to study the transformation properties of the spinors... 

We are not going to review here the transformation properties of spatial wave functions under the symmetry operations of molecular point groups. To prepare the discussion of the transformation properties of spinors, we shall put some effort, however, in discussing the symmetry operations of 0(3)+, the group of proper rotations in 3D coordinate space (i.e., orthogonal transformations with determinant + 1). Reflections and improper rotations (orthogonal transformations with determinant -1) will be dealt with later. 

Then, we can benefit largely from factors that become zero upon resolution of an expectation value over two Slater determinants because of the orthogonality of two different spinors. Also, the Slater determinants turn out to be orthonormal then. 

By contrast to the numerical MCSCF method discussed in the last chapter, the basis-set approach has the convenient advantage that the virtual orbitals come for free by solution of the Roothaan equation. While the fully numerical approaches of chapter 8 do not produce virtual orbitals, as the SCF equations are solved directly for occupied orbitals only and smart b)q)asses must be devised, this problem does not show up in basis-set approaches. Out of the m basis functions, only N with N matrix Fock operator produces a full set of m orthogonal molecular spinor vectors that can be efficiently employed in the excitation process of any Cl-like method. 

The two-component spinor functions cpfi can be chosen as spin orbitals which are the direct product of real scalar functions with spin functions, A (x) a,jS. Such a choice reduces the cost of basis orthogonalization since only real matrices are involved and different spins are decoupled. 

Many-electron wave functions correct to oi may be expanded in a set of CSFs that spans the entire N-electron positive-energy space j (7/J 7r), constructed in terms of Dirac one-electron spinors. Individual CSFs are eigenfimctions of the total angular momentum and parity operators and are linear combinations of antisymmetrized products of positive-energy spinors (g D(+ ). The one-electron spinors are mutually orthogonal so the CSFs / (7/J 7r) are mutually orthogonal. The un-... 

In all-electron calculations, the number of radial nodes of an atomic orbital (AO) increases by one as the principal quantum increases by one. Accordingly, while a Is atomic orbital is nodeless (in this and the following discussion, nodes at r = 0 and are ignored), the 2s, 3s, 4s, and higher s orbitals contain one, two, three, and so forth radial nodes. Radial nodes are required to ensure that the radial portions of the atomic wavefunctions remain orthogonal. With replacement of core electrons and orbitals by a potential, one must remove the appropriate number of nodes in the valence orbitals to ensure that the s, p, d, f, etc., orbital with the lowest principal quantum number not replaced by the ECP is nodeless, as is the Is, 2p, 3d, 4f, etc., atomic orbital in an all-electron calculation. Wavefunctions derived from relativistic calculations should be referred to as spinors (to denote their j dependence, j = V2).i We will use the terms spinor and orbital interchangeably. 

Once the projection operators have been introduced we may remove the requirement that the valence spinors should be orthogonal to the core spinors From the properties of determinants, we know that we ean always add a linear combination of the core spinors to the valence spinors without ehanging the total wave function. The resulting spinor we term a pseudospinor,... 

This pseudospinor is normalized on the metric G and is orthogonal to other pseudospinors on the same metric. The coefficients can be determined to minimize the density of the pseudospinor in the core region. For this reason, we will call the sum of core spinors the core tail. 

The generalized Philips-Kleinman pseudopotential depends on the eigenvalue of the spinor v, unlike the frozen-core pseudopotential, which depends only on the core spinors. The appearance of the term in the pseudopotential came about because we transferred a term from right-hand side of (20.21). This means that we have changed the metric, which has implications for orthogonality that we pursue later. The new operator makes the core spinors degenerate with the valence spinor ... 

In parallel with the development of pseudopotentials, a second approach to the frozen-core model has been taken. In this approach, the core orthogonality is retained, and the model includes an explicit representation of the core spinors. The rationale for retaining the core tail in the valence functions is that the cost of the primitive integrals is less important than the cost of retaining all the core basis functions. Some approximations to the core tails are generally made because the integral cost is still significant. 

If the core orthogonality is retained, there is no necessity to insert the projection operators around the core Fock operator in (20.5), but we must still ensure that core spinors are kept out of the valence space. In an atom it is easy to maintain the orthogonality, but in a molecule the basis functions on another center expand into a linear combination of functions on the frozen-core center, including core spinors. 

For an arbitrary spinor r, we can expand the spinor into a contribution from the pseudospinor v and an orthogonal term. 

In the third and most general case, where the valence pseudospinor is not orthogonal to the core or to the trial function, the gradient is again given by (20.103). For a core spinor ft, we can use the expansion of the valence pseudospinor in terms of the complete set of spinors... 

This expression contains a core projector matrix element in addition to the model potential Fock matrix element. However, because it is off-diagonal the core projector term could be of either sign, depending on the contribution of the orthogonal component to the trial spinor. Whatever contributions to the Fock matrix elements come from core spinors, their signs are changed by the projector matrix element. In any case, the... 

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