Spin-free operators

Let us first consider one-electron operators. Following the general discussion in Section 1.4.1, a spin-free one-electron operator of the form 

The integrals entering the second-quantization operator / vanish for c posite spins since the first-quantization operator is spin-free  

The second-quantization representation of the spin-free one-electron operator (2.2.1) now becomes 

We now turn our attention to spinless two-electron operators. According to the discussion in Section 1.4.2, the second-quantization representation of a general spin-free two-electron operator of the fcam 

Most of the terms in this operator vanish because of the orthogonality of the spin functions 

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But this is not the full story. The Hamiltonian operator employed is a spin-free operator and does not work on the spin functions a and p. H commutes therefore with the spin operators Sz and S ... 

Most Hamiltonians of physical interest are spin-free. Then the matrix elements in Eq. (9) depend only on the space part of the spin orbitals and vanish for different spin by integration over the spin part. Then it is recommended to eliminate the spin and to deal with spin-free operators only. We start with a basis of spin-free orbitals cpp, from which we construct the spin orbitals excitation operators carry orbital labels (capital letters) and spin labels... 

The quantum mechanical operators can be divided according to spatial and spin properties. In first quantization a pure spatial (spin free) operator Fc does not change the spin functions so Fc commutes with the spin function... 

Any nonzero volume piece pd(r) of the nondegenerate ground state electron density fully determines the ground state energy E, the ground state wavefunction P (up to a phase factor), and the expectation values of all spin-free operators defined by the ground state wavefunction VP . 

Energy is not the only property that is so determined by the electron density fragment pc. Since the (non-degenerate, ground state) local electron density pc(r) in any standard domain c fully determines the complete density pit), which in turn fully determines the molecular wavefunction P (up to a phase factor), all molecular properties P which can be expressed as expectation values of spin-free operators defined by the ground state wavefunction P are also determined by the local electron density pc(r) in the standard domain c. Consequently, any such property P is also a unique functional of the local electron density pc(r) within the standard domain c ... 

The commutator expressions, Eqs (120) and (121), are sufficient to derive expressions for the matrix elements required for the MCSCF optimization process. This results from the fact that both the orbital transformation and the Hamiltonian operator are written in terms of the generators and generator products of Eq. (117). Since all of these operators involve explicit references only to the spatial orbitals, and not to the spin orbitals, it would be possible to eliminate reference to the spin-orbitals entirely if the expansion kets could be represented in such a spin-free method and if matrix elements of these spin-free operators could be calculated without reference to the spin orbitals. 

That is, a piece of the ground-state electron density fully determines the ground state electron density of the entire system. As also follows from the Hohenberg-Kohn theorem, the ground-state energy E, and the ground-state wavefunction P are uniquely determined by p (r ), hence they are also uniquely determined by the electron density p r ) of the subsystem d. Consequently, the expectation values of all spin-free operators defined by the ground-state wavefunction are uniquely determined by the electron density p /,r ) of any nonzero volume subsystem d. 

The gauge term, which is the difference between the Gaunt interaction and the Breit interaction, produces a spin-free operator that can be interpreted as an orbit-orbit interaction. Thus, both the Gaunt interaction and the gauge term of the Breit interaction give rise to spin-free contributions to the modified Dirac operator. We will use the developments of this section in chapter 17 to derive the Breit-Pauli Hamiltonian. 

The first class contains the spin-free operators. 

The expectation value of a general one-electron but spin-free operator O = X) d i) in the unperturbed ground state is given as... 

In extension of Eq. (3.46) we can then express the first-order correction to a field-dependent expectation value of the general but spin-free operator O with the first-order... 

All approximate ab initio methods presented in Chapter 9 are based on Slater determinants built with molecular orbitals. In this section we will therefore derive expressions for the expectation value ( o O o) of a general one-electron but spin-free operator O = o(0 an approximate closed-shell wavefunction o) in terms of the... 

In addition to these second-order corrections to the RPA matrices there are three new matrices due to the operators. In the following, we present explicit expressions for them in terms of spatial orbitals (f>p and for two spin-free operators Pa and using a biorthogonal set of double excitation operators qq (Bak et ai, 2000). 

The nonrelativistic Hamiltonian (2.2.18) is a spin-free operator - that is, a spin tensor operator of zero rank see Section 2.3. Determinants of different spin projections therefore give vanishing Hamiltonian matrix elements and we may restrict the determinants of the Cl expansion to have the same spin projection. If the total number of electrons is N and the spin projection is M. the numbers of electrons with alpha and beta spins are given by... 

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