Exponential unitary operator

D. Siminovitch, T. S. Untidt and N. C. Nielsen, Exact effective Hamiltonian theory. II Expansion of matrix functions and entangled unitary exponential operators. J. Chem. Phys., 2004, 120, 51-66. 

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... 

If we have chosen a basis where the Hamiltonian matrix is diagonal, the terms that create and annihilate pairs vanish, because the matrix elements for these terms contain one electron and one positron index. We are left with an operator that conserves particle number. It is in this basis that the interpretation of particles is made. A basis change in Fock space may be effected by the unitary exponential operator (see for example Helgaker et al. 2000)... 

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... 

For variational methods, such as Hartree-Fock (HF), multi-configurational self-consistent field (MCSCF), and Kohn-Sham density functional theory (KS-DFT), the initial values of the parameters are equal to zero and 0) thus corresponds to the reference state in the absence of the perturbation. The A operators are the non-redundant state-transfer or orbital-transfer operators, and carries no time-dependence (the sole time-dependence lies in the complex A parameters). Furthermore, the operator A (t)A is anti-Hermitian, and tlie exponential operator is thus explicitly unitary so that the norm of the reference state is preserved. Perturbation theory is invoked in order to solve for the time-dependence of the parameters, and we expand the parameters in orders of the perturbation... 

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... 

For constant or slowly varying Hamiltonians, therefore, the unitary exponential propagate seems excellent A major problem, however, is that the representation of the exponential of an operator may be difficult Tal-Ezer and Kosloffi S for example, used a Chebyshev... 

An alternative relax-and-drive procedure can be based on a strictly unitary treatment where the advance from Iq to t is done with a norm-conserving propagation such as provided by the split-operator propagation technique.(49, 50) This however is more laborious, and although it conserves the norm of the density matrix, it is not necessarily more accurate because of possible inaccuracies in the individual (complex) density matrix elements. It can however be used to advantage when the dimension of the density matrix is small and exponentiation of matrices can be easily done.(51, 52)... 

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. 

The orthogonal complement set of states to I 0> may be obtained operating with the exponential unitary operator on the orthogonal complement set of states. In Exercise 14 it is shown... 

These operators are used to define the isomorphism p(g) which characterizes the unitary irreducible and infinite-dimensional representation of H", through the exponential map... 

In the above expression 0) is the reference determinant, whereas exp[—ic(t)] is the (unitary) orbital-rotation operator, being the exponential of the anti-Hermitian operator... 

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 ... 

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). 

Lua supports the usual array of arithmetic operators + for addition, - for subtraction, for multiplication, / for division, % for modulo, for exponentiation and unitary - before any number. The supported relational operators are ... 

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. 

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