Second Unitary Transformation

After the initial transformation Uq, the new odd part Op in fx needs to be deleted by the next transformation Ux, and all that we require is that new odd terms are of higher order in the scalar potential V. Now, we do not restrict any of the subsequent unitary transformations to any specific choice of unitary transformation and employ the most general form of Eq. (11.57) without specifying the expansion coefficients that satisfy the xmitarity conditions Eqs. (11.60)-(11.66). Because the first term of each expansion of the imitaiy matrices according to Eq. (11.57) is the unit operator 1, a transformation of any operator yields exactly this operator plus correction terms. Hence, the first free-particle transformation already produced the final zeroth- and first-order even terms. So, we already have all even terms up to 1st order for the DKH Hamiltonian collected, 

For later convenience the odd and antihermitean expansion parameter is denoted by W/ instead of W, and shall be of exactly i-th order in V. 

Note that So, S, and are independent of W[ and therefore completely determined from the very beginning. Again, the subscript attached to each term of the Hamiltonian denotes its order in the external potential, whereas the superscript in parentheses indicates that such a term belongs to the intermediate, partially transformed Hamiltonian relevant only for the higher-order terms to be finalized by subsequent transformations. Only those even terms which will not be affected by the succeeding unitary transformations Lfj, (f = 2,3.) carry no superscript. 

Wj is chosen to guarantee O = 0, and therefore the following condition for Wj is obtained. 

It is clear that is then an odd operator. As a consequence, this operator alone cannot be represented in any truly two-component theory, but only its products with other odd operators which are then even and hence representable in a two-component theory. 

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Now we find the covalent (dynamic) contribution to the AOM parameters elxx,. Inverting the relation between the matrix of the contribution vA to the crystal field and the matrix of the AOM parameters eA - this is the second unitary transformation of the two mentioned in the beginning of this section - we get ... 

The fact that the term of order s3 in Eq. (144) is off-block-diagonal implies that if we perform a second unitary transformation ee W3, there will be no term of order s3 in the diagonal block projection Ih, and thus the next order correction for the diagonal block, and therefore for eigenvalues, will be of order 4 (given by... 

We will now investigate how the second-order energy (T) at a particular point T = T(R) changes if T undergoes a small variation. Such a change can be described by multiplying U with a second unitary transformation U(AR) ... 

The unitary transformation U can be determined as the product of two ttansfor-mations f/of/i- The first transformation f/o is the free-particle Fouldy-Wouthuysen transformation and leads to the approximate separation of the electronic and positronic spectra = U HqUq. The second unitary transformation is... 

This result holds equally well, of course, when R happens to be the operator representing the entropy of an ensemble. Both Tr Wx In Wx and Tr WN In WN are invariant under unitary transformations, and so have no time dependence arising from the Schrodinger equation. This implies a paradox with the second law of thermodynamics in that apparently no increase in entropy can occur in an equilibrium isolated system. This paradox has been resolved by observing that no real laboratory system can in fact be conceived in which the hamiltonian is truly independent of time the uncertainty principle allows virtual fluctuations of the hamiltonian with time at all boundaries that are used to define the configuration and isolate the system, and it is easy to prove that such fluctuations necessarily increase the entropy.30... 

On matrix form the non-unitary transformations (27) and (30) of the previous section are easily extended to the complete Hamiltonian and have therefore allowed relativistic and non-relativistic spin-free calculations of spectroscopic constants and first-order properties at the four-component level (see, for instance. Refs. [45 7]). In this section, we consider the elimination of spin-orbit interaction in four-component calculations of second-order electric and magnetic properties. Formulas are restricted to the Hartree-Fock [48] or Kohn-Sham [49] level of theory, but are straightforwardly generalized. 

Importantly, the anti-Hermitian CSE may be evaluated through second order of a renormalized perturbation theory even when the cumulant 3-RDM is neglected in the reconstruction. The anti-Hermitian part of the CSE [27, 31, 63] is the stationary condition for two-body unitary transformations of the A-particle wave-function [31, 32], and hence the two-body unitary transformations may easily be evaluated with the anti-Hermitian CSE and RDM reconstruction without the many-electron Schrodinger equation. The contracted Schrodinger equation in conjunction with the concepts of reconstruction and purification provides a new, important approach to computing the 2-RDM directly without the many-electron wavefunction. 

Because polynomials of orbital occupation number operators do not depend on the off-diagonal elements of the reduced density matrix, it is important to generalize the Q, R) constraints to off-diagonal elements. This can be done in two ways. First, because the Q, R) conditions must hold in any orbital basis, unitary transformations of the orbital basis set can be used to constrain the off-diagonal elements of the density matrix. (See Eq. (56) and the surrounding discussion.) Second, one can replace the number operators by creation and annihilation operators on different orbitals according to the rule... 

Warnings (i) The Tpq do not form a second-rank tensor and so unitary transformations must be carried out using the four-index notation Tijki. (ii) The contraction of TiJki may be accompanied by the introduction of numerical factors, for example when 7(4) is the elastic stiffness (Nye (1957)). 

The second method, due to King et al,76 is well illustrated by considering the overlap matrix Tab between two Slater determinants Da, Db. These two determinants remain unaltered by unitary transformations U and V of the sets of spin orbitals which constitute Da and Db, respectively, while in the new representation rAB becomes UTABV. One may now choose these transformations so as to diagonalize rAB, giving... 

Whitten and Pakkanen proposed a so-called localization scheme in which the interaction of the adsorbate levels with the large number of metal levels can be described by those orbitals for which the interaction < Htf t> is large. A unitary transformation can be introduced such that <4>k I Heff I 4t> I is maximized. This procedure yields one set of functions which physically represent orbitals localized on the designated surface atoms, bonds between these atoms, as well as bonds linking the designated atoms with the remainder of the cluster. A second set of functions consists of so-called interior orbitals. They are treated as an invariant core in each configuration for the step of the configuration... 

In this subsection we will combine the general ideas of the iterative perturbation algorithms by unitary transformations and the rotating wave transformation, to construct effective models. We first show that the preceding KAM iterative perturbation algorithms allow us to partition at a desired order operators in orthogonal Hilbert subspaces. Its relation with the standard adiabatic elimination is proved for the second order. We next apply this partitioning technique combined with RWT to construct effective dressed Hamiltonians from the Floquet Hamiltonian. This is illustrated in the next two Sections III.E and III.F for two-photon resonant processes in atoms and molecules. 

We remark that, as opposed to Eqs. (145)—(129), in this construction two unitary transformations are needed to obtain the effective eigenvectors to second order (after the first transformation, where keeping only the diagonal blocks in Eq. (164) yields the eigenvalues to second order, but not the eigenvectors). 

Secondly, the commutator is the Lie product33 of the operators X Xs and Xu this choice of multiplication is particularly appropriate when one realizes that the X XS are the generators of the semisimple compact Lie group U , which is associated with the infinitesimal unitary transformations of the Euclidean vector space R (e.g., the space of the creation operators).34 With the preceding comments, the action of the transformation operator on the creation operators can formally be written in the usual form of the transformation law for covariant vectors,33... 

As proposed by Marzari and Vanderbilt [219], an intuitive solution to the problem of the non-uniqueness of the unitary transformed orbitals is to require that the total spread of the localized function should be minimal. The Marzari-Vanderbilt scheme is based on recent advances in the formulation of a theory of electronic polarization [220, 221]. By analyzing quantities such as changes in the spread (second moment) or the location of the center of charge of the MLWFs, it is possible to learn about the chemical nature of a given system. In particular the charge centers of the MLWFs are of interest, as they provide a classical correspondence to the location of an electron or electron pair. 

The optimization method outlined in Sections II.C-E shows very fast and stable convergence behaviour when applied to CASSCF wavefunctions, in which all orbital rotations between occupied orbitals are redundant. However, experience has shown that convergence is often much slower when orbital rotations between strongly occupied valence orbitals have to be optimized. It has been shown that this is due to the fact that the energy approximation (T) is not invariant with respect to a unitary transformation between two doubly occupied orbitals . If the 2 x 2 transformation in Eq. (34) is applied to a single Slater determinant with just two doubly occupied orbitals, the second-... 

The second major method leading to two-component regular Hamiltonians is based on the Douglas-Kroll transformation (Douglas and Kroll 1974 Hess 1986 Jansen and Hess 1989). The classical derivation makes use of two successive unitary transformations... 

In the simple CV(2)-DFT theory [144], the unitary transformation of (33) is carried out to second order in U. We thus obtain the occupied excited state orbitals to second order as... 

The second possibility to reduce the four-component Dirac spinor to two-component Pauli form is to decouple the Dirac equation, i.e., to transform the Dirac Hamiltonian to block-diagonal form by a suitably chosen unitary transformation U,... 

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