Unitary exponential parametrization

Table 14.1 Number of matrix multiplications P(n) required for the evaluation of DKHn Hamiltonians with an exponential parametrization of the unitary transformation according to Ref. [647],...

The exponential parametrization of a unitary operator is independent in the sense that there are no restrictions on the allowed values of the numerical parameters in the operator - any choice of numerical parameters gives rise to a bona fide unitary operator. In many situations, however, we would like to carry out restricted spin-orbital and orbital rotations in order to preserve, for example, the spin symmetries of the electronic state. Such constrained transformations are also considered in this chapter, which contains an analysis of the symmetry properties of unitary orbital-rotation operators in second quantization. We begin, however, our exposition of spin-orbital and orbital rotations in second quantization with a discussion of unitary matrices and matrix exponentials. 

The exponential parametrization of a unitary matrix in (3.1.9) is a general one, applicable under all circumstances. We shall now consider more special forms of unitary matrices. We begin by writing the anti-Hermitian matrix X in the form... 

Chapters 1-3 introduce second quantization, emphasizing those aspects of the theory that are useful for molecular electronic-structure theory. In Chapter 1, second quantization is introduced in the spin-orbital basis, and we show how first-quantization operators and states are represented in the language of second quantization. Next, in Chapter 2, we make spin adaptations of such operators and states, introducing spin tensor operators and configuration state functions. Finally, in Chapter 3, we discuss unitary transformations and, in particular, their nonredundant formulation in terms of exponentials of matrices and operators. Of particular importance is the exponential parametrization of unitary orbital transformations, used in the subsequent chapters of the book. 

At this stage it is, however, not clear which of these parametrizations should be preferred over the others for application in decoupling procedures and whether they all yield identical block-diagonal Hamiltonians. Furthermore, the four possibilities given above to parametrize unitary transformations differ obviously in their radius of convergence Rc, which is equal to unity for the square root and the McWeeny form, whereas Rc = 2 for the Cayley parametrization, and Rc = co for the exponential form. 

Most of the theory can be developed using only the one-electron part of this operator, with rather straightforward extensions to include the two-electron part. To effect the rotations in the function space we employ the exponential rotation opaator U = exp(iX), introduced in (5.35), but parametrized in terms of the operator k = iX. We want the rotations to preserve orthonormality in the set of one-particle functions, and therefore require that U he a unitary operator, that is... 

Our parametrization of the MCSCF state is a straightforward combination of the parametrizations of the Hartree-Fock state in Chapter 10 and of the Cl state in Chapter 11. The orbital variations are carried out by means of an exponential unitary operator and the variations in the configuration space are expressed by means of a configuration vector added orthogonally to the reference state ... 

We have now satisfied the first of the three requirements for the unitary parametrization stated in the introduction to Section 3.1. To satisfy the second requirement, we note that, for any anti-Hermitian matrix X, the exponential exp(X) is always unitary since, from the relation X = —X, it follows that... 

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