Natural spinors

M. Klobukowski. Two-component natural spinors from two-step spin-orbit coupled wave functions. /. Chem. Phys., 134 (2011) 214107. 

Quiney (2000)]. The quadratically convergent NR algorithm for relativistic MCDF SCF calculations has been discussed in detail in previous work [Vilkas et al. (1998a)] and is not discussed further. To remove the arbitrariness of the MC SCF spinors and density weighting, the canonical SCF spinors are transformed into natural spinors for subsequent pertur-... 

CSF 1( 1 Jn). Jp and Kp are the usual Coulomb and exchange operators constructed in natural spinors. 

While seminal works intended to reveal SOC effects on the bonding schemes were discussed in term of spinors [9-13], canonical molecular spinors are not suited for the bonding analysis in complex systems, as opposed to small and/or symmetric model systems. Some have promoted the use of localized spinors [14], and in order to recover some chemical significance in terms of bonding, lone pairs and core orbitals, natural spinors similar to natural orbitals in the non-relativistic firameworks have been derived and implemented [15, 16]. It is worth noting that the concept of bond order in the context of multiconfigurational wave functions have been extended recently to two-step spin-orbit coupling approaches [17]. 

For spin-orbit coupling Cl wave functions, two-component natural orbitals (natural spinors) maybe formed and analyzed (Zeng et al. 2011a, b)... 

Section VI shows the power of the modulus-phase formalism and is included in this chapter partly for methodological purposes. In this formalism, the equations of continuity and the Hamilton-Jacobi equations can be naturally derived in both the nonrelativistic and the relativistic (Dirac) theories of the electron. It is shown that in the four-component (spinor) theory of electrons, the two exha components in the spinor wave function will have only a minor effect on the topological phase, provided certain conditions are met (nearly nonrelativistic velocities and external fields that are not excessively large). 

This is the most general expression obtained from a set of natural spin orbitals written in spinor form as... 

As a characteristic feature, both the gap functions have nodes at poles (9 = 0,7r) and take the maximal values at the vicinity of equator (9 = 7t/2), keeping the relation, A > A+. This feature is very similar to 3P pairing in liquid 3He or nuclear matter [17, 18] actually we can see our pairing function Eq. (39) to exhibit an effective P wave nature by a genuine relativistic effect by the Dirac spinors. Accordingly the quasi-particle distribution is diffused (see Fig. 3)... 

The spin density should follow from the density matrix (38), which includes the spin variables. As in (42), Qa(x x ) will be a sum of terms containing the various spinor components, summed over all spin-orbitals in the natural expansion. A typical term will be... 

For the electro-nuclear model, it is the charge the only homogeneous element between electron and nuclear states. The electronic part corresponds to fermion states, each one represented by a 2-spinor and a space part. Thus, it has always been natural to use the Coulomb Hamiltonian Hc(q,Q) as an entity to work with. The operator includes the electronic kinetic energy (Ke) and all electrostatic interaction operators (Vee + VeN + Vnn)- In fact this is a key operator for describing molecular physics events [1-3]. Let us consider the electronic space problem first exact solutions exist for this problem the wavefunctions are defined as /(q) do not mix up these functions with the previous electro-nuclear wavefunctions. At this level. He and S (total electronic spin operator) commute the spin operator appears in the kinematic operator V and H commute with the total angular momentum J=L+S in the I-ffame L is the total orbital angular momentum, the system is referred to a unique origin. 

All electrons, protons and neutrons, the elementary constituents of atoms, are fermions and therefore intrinsically endowed with an amount h/2 of angular momentum, known as spin. Like mass and charge, the other properties of fermions, the nature of spin is poorly understood. In quantum theory spin is treated purely mathematically in terms of operators and spinors, without physical connotation. 

The DC-CASCI (12, 4) calculation was performed to construct reference functions. The active space includes the molecular spinors, which have atomic nature of 6s1/2, 6p1/2, 6py2 of T1 and 1 s1/2 of H, and two virtual molecular spinors. The DC-CASPT2 (10, 12, 110) calculation followed and this choice of active space provided smooth potential curves for four low-lying states of T1H at the DC-CASPT2 level. 

According to general properties of Pauli matrices (a p)2 = p2 hence (9) is recognized as Schrodinger s equation, with E and p in operator form. On defining the electronic wave functions as spinors both Dirac s and Schrodinger s equations are therefore obtained as the differential equation describing respectively non-relativistic and relativistic motion of an electron with spin, which appears naturally. 

A significant point here is that it is not the squared invariant ds2 that is to underlie the covariance of the laws of nature. It is rather the linear invariant ds that plays this role. How, then, do we proceed from the squared metric to the linear metric That is to say, how does one take the square root of ds2l The answer can be seen in Dirac s procedure, when he factorized the Klein-Gordon equation to yield the spinor form of the electron equation in wave mechanics -the Dirac equation. Indeed, Dirac s result indicated that by properly taking the square root of ds2 in relativity theory, extra spin degrees of freedom are revealed that were previously masked. 

For particles with spin-1/2 we would expect (on the basis of nonrelativistic quantum mechanics) that spinors with two components would be sufficient. But the Dirac spinors have to be (at least) four-dimensional. A mathematical reason lies in the nature of the algebraic properties that have to be satisfied by the Dirac matrices a and 0 if the Dirac equation should satisfy the relativistic energy-momentum relation in the sense described above, see (6). 

While the angular parts (including spin) of the spinors emerge naturally from the spherical symmetry of the problem, it is slightly more complicated to find the radial parts P(r) and Q r). For the case of a point nucleus, these can be shown to take the form of the product of an exponential function, a power of r and a polynomial,... 

The large component accounts for most of the electron density of a spinor, and as such will carry the largest weight in basis set optimizations. It also has the larger amplitude, and as such must weigh heavily in any fitting scheme. This is only natural, and for most purposes, including standard chemical applications, creates no problems. However, there are some properties that depend heavily on the quality of the small component description. One of these would be the interaction of a possible electric dipole moment, dg of the electron with an applied external field, S. This interaction is described by the operator [20]... 

Since even and odd operators obey the same multiplication rules as natural numbers, i.e., even times odd is odd, etc., this is obviously an odd and antihermitean operator of second order in the external potential, which is linear in af /af. W2 is thus a second-order integral operator in momentum space, whose action on a spinor 0 is defined by... 

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