Spin-free excitation operators

C. Spin-Free Excitation Operators and -Particle Density Matrices... 

Greek letters). We define spin-free excitation operators carrying only orbital labels, by summation over spin... 

For a function ip with a given spin-multiplicity (25 - -1), we shall consider all the CSFs with Mj = (25 -1-1), generated from some base CSF (f>g with inactive occupied orbitals doubly filled, some active orbitals doubly filled, a set of active orbitals with up-spin, a set tig orbitals with down-spin, coupled to a singlet, and another set active orbitals, all with up-spin such that Mg = (25 - - 1). With this generation scheme, every model CSF can be written as some spin-free excitation operator acting on the base function tpg. 

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 spin-free two-particle excitation operators and density matrices are symmetric with respect to simultaneous exchange of the upper and lower indices, but neither symmetric nor antisymmetric with respect to exchange of either upper or lower indices separately ... 

For spin-free fe-particle excitation operators and density matrices, linear combinations that transform as irrep of the symmetric group Sk can be defined in an analogous way [15]. 

Because the convenience of the one-electron formalism is retained, DFT methods can easily take into account the scalar relativistic effects and spin-orbit effects, via either perturbation or variational methods. The retention of the one-electron picture provides a convenient means of analyzing the effects of relativity on specific orbitals of a molecule. Spin-unrestricted Hartree-Fock (UHF) calculations usually suffer from spin contamination, particularly in systems that have low-lying excited states (such as metal-containing systems). By contrast, in spin-unrestricted Kohn-Sham (UKS) DFT calculations the spin-contamination problem is generally less significant for many open-shell systems (39). For example, for transition metal methyl complexes, the deviation of the calculated UKS expectation values S (S = spin angular momentum operator) from the contamination-free theoretical values are all less than 5% (32). 

Following the customary terminology, we will call inactive holes the inactive occupied orbitals, doubly filled in every model CSF. The inactive particles will refer to aU the orbitals unoccupied in every CSF. Orbitals which are occupied in some (singly or doubly) but unoccupied in others are the active orbitals. In our spin-free form, the labels are for orbitals only, and not for spin orbitals. From the mode of definition, no active orbital can be doubly occupied in every model CSF. We want to express the cluster operator T, inducing excitations to the virtual functions, in terms of excitations of minimum excitation rank, and at the same time wish to represent them in a manifestly spin-free form. To accomplish this, we take as the vacuum—for excitations out of 4> — the largest closed-shell portion of it, For each such vacuum, we redefine the holes and particles, respectively, as ones which are doubly occupied and unoccupied in < 0 a-The holes are denoted by the labels. .., etc. and the particle orbitals are denoted as a, etc. The particle orbitals are totally unoccupied in any or are necessarily... 

It is worth to mention that in the UHF formalism, due to the orthogonality of the spin functions, it is not allowed to excite an electron occupying a spin-orbital to the / spin-orbital (and vice versa). As it was already mentioned in Subsection 3.2, in the case of explicitly correlated CCSD theory, the cluster operator is supplemented with the additional excitation operator T2/ [Eq. (40)]. This operator is responsible for the explicitly correlated treatment and involves additional excitations (F12 excitations) into the complementary basis. In the spin-free formalism its mathematical form was already shown and briefly discussed [Eq. (40)], in the spin-orbital formalism this operator can be introduced as... 

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 incorporation of spin in second quantization leads to operators with different spin synunetry properties as demonstrated in Section 2.2. Thus, spin-free interactions are represented by operatOTs that are totally symmetric in spin space and thus expressed in terms of orbital excitation operators that affect alpha and beta electrons equally, whereas pure spin interactions are represented by excitation operators that affect alpha and beta electrons differently. For the efficient and transparent manipulation of these operators, we shall apply the standard machinery of group theory. More specifically, we shall adopt the theory of tensor operators for angular momentum in quantum mechanics and develop a useful set of tools for the construction and classification of states and operators with definite spin symmetry properties. 

The spin-free one- and two-electron operators (2.2.6) and (2.2.15) are thus expressed entirely in terms of the singlet excitation operator and we may, for example, write the electronic Hamiltonian (2.2.18) in the form... 

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