Unitary exponential operator transformation

Consider a trial Kohn-Sham (KS) determinant, either a closed shell determinant or an open-shell high-spin determinant where all singly occupied orbitals have a spin. It is parameterized by a real unitary exponential operator, and the purpose of the transformation is to transform the orbitals to a state of minimum energy... 

Rotations are likewise unitary transformations, and we shall see that they can also be represented by an exponential operator. Let D(a) be a rotation about the z-axis, so that... 

This shows that the effect of an orbital transformation, as in Eq. (76), on any expansion ket is exactly represented by the action of the exponential operator exp( - iA) on that ket. The important feature of the operator representaton is that it gives the effect of the orbital transformation expressed in the original orbital basis. Since Eq. (99) is valid for any ket, it also holds for any linear combination of kets and, therefore, for the MCSCF wavefunction. It may be noted that when A = 0, this unitary operator reduces to the identity operator. For small values of the parameters A , it is useful to consider the truncated expansion of the exponential operator... 

In the preceding sections, the occupation vectors were defined by the occupation of the basis orbitals jJ>. In many cases it is necessary to study occupation number vectors where the occupation numbers refer to a set of orbitals cfe, that can be obtained from by a unitary transformation. This is, for example, the case when optimizing the orbitals for a single or a multiconfiguration state. The unitary transformation of the orbitals is obtained by introducing operators that carry out orbital transformations when working on the occupation number vectors. We will use the theory of exponential mapping to develop operators that parameterizes the orbital rotations such that i) all sets of orthonormal orbitals can be reached, ii) only orthonormal sets can be reached and iii) the parameters are independent variables. 

As already noted, there are many similarities between the exponential unitary transformations in configuration space and in orbital space. Comparing with the results for orbital transformations in Chapter 3, we note that the operators S and k are both anti-Hermitian, as are the matrices S and x. Moreover, whereas exp(—S)]P) represents a unitarily transformed configuration state, exp(—ic) P) represents a state where the spin orbitals have been unitarily transformed as exp(—x)ap exp(ic). 

The Fock space as introduced in Chapter I is defined in terms of a set of orthonormal spin orbitals. In many situations - for example, during the optimization of an electronic state or in the calculation of the response of an electronic state to an external perturbation - it becomes necessary to carry out transformations between different sets of orthonormal spin orbitals. In this chapter, we consider the unitary transformations of creation and annihilation operators and of Fock-space states that are generated by such transformations of the underlying spin-orbital basis. In particular, we shall see how, in second quantization, the unitary transformations can be conveniently carried out by the exponential of an anti-Hermitian operator, written as a linear combination of excitation operators. 

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. 

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. 

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