Unitary group generators

Unitary-group generators have in fact already been encountered in Section 8.2, where they were represented as operators in Fock space describing the mappings of a spin-orbital basis, and, although we shall use a spin-free formalism, it is also useful to see how they occur in the second-quantization approach. In (8.2.18) et seq. it was established that the operator... 

Table 10.1 Non-zero matrix elements of unitary-group generators between...

We have used the unitary group generators, for conciseness. 

The E operators are the elementary operators, the generators, in unitary group theory. Below we give some commutator relations between E operators, e operators, creation and annihilation operators that are important for developing a good strategy for evaluating many of the quantities that show up in the later derivation. 

The indices i and j in (3 7) now refer to the n molecular orbitals without the spin factor.These operators fulfill the same commutator relation as the generators of the unitary group of dimension n, and are often referred to as generators. The commutator relation has the following form ... 

We will finish this section by pointing out a property which will be useful later on in this chapter. In the mathematical lectures the generators of the unitary group have been defined as,... 

In order to be able to write out all the terms of the direct Cl equations explicitly, the Hamiltonian operator is needed in a form where the integrals appear. This is done using the language of second quantization, which has been reviewed in the mathematical lectures. Since, in the MR-CI method, we will generally work with spin-adapted configurations a particularly useful form of the Hamiltonian is obtained in terms of the generators of the unitary group. The Hamiltonian in terms of these operators is written,... 

This expression is often called the resolution of the identity . The expression (6.6) for the two-electron coupling coefficient can now be rewritten by introducing expression (7.6) between the generators of the unitary group. The result is,... 

The second approximate scheme we will discuss here is the internally contracted Cl (ICCI) method. In this method correlating configurations are formed by applying excitation operators (the generators of the unitary group) directly on the full reference Cl vector. The four types of configurations thus formed can be written as,... 

The relevant Casimir operator of unitary group U21+1 follows from (5.33) if we include in it the term with k = 0. The Casimir operator of symplectic group Sp4i+2, whose generators are tensors Uk and Vk+l with the odd sums of ranks, may be defined in the following way ... 

The Casimir operator for so(3) commutes with all generators and is given by J2 = J + J + J3. The unitary irreducible representations are characterized by a single quantum number j = 0,1, ,..., which also includes the spin representations of su(2) corresponding to the special unitary group SU(2) when j = j,, and are given by... 

It is clear that the new generation of computers - supercomputers when used in conjunction with theoretical developments such as the linked diagram theory or the unitary group theory, is going to have a significant impact on the accuracy attainable in quantum chemistry and on the size of problem which may be treated. 

Graphical Methods of Spin Algebras. In most problems of chemical interest, the molecular hamiltonian can be taken to be spin-free and can be written172 173 in terms of the generators of the unitary group, U(n), where n is the number of basis functions... 

For spin-orbitals, the creation and annihilation operator pair is a generator of the unitary group, and so Eq. (52) can be written... 

Eq. (120) is the same commutation relation as is satisfied by the generators of the unitary group U(n), the group of all n-dimensional unitary matrices. For this reason, the operators are often referred to as generators " . The operator rank reductions shown in Eqs (120) and (121) prove to be important in some of the following derivations. 

The unitary group approach attempts to solve this redundancy in two ways. First, the expansion terms are constructed in such a way that only spin eigenfunctions of the correct S value are considered. Secondly, the generator and generator product matrix elements in the CSF expansion space are computed in such a way that no reference to determinants is made, either implicitly or explicitly. This leads to computational schemes that depend only... 

The subset of operators satisfying Eq. (232) is usually known before the MCSCF optimization process is initiated. In fact, many CSF generation methods, including the graphical unitary group approach, allow the specification of the operators of Eq. (232) or the orbital subsets that define these rotation operators. The corresponding orbital rotation parameters may then be deleted from the MCSCF optimization process for any state and for any geometry for wavefunctions expanded in this CSF space. The direct product... 

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