Unitary transformation configuration

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... 

The MO (molecular orbitals) of a polyatomic system are one-electron wave-function k which can be used as a (more or less successful) result for constructing the many-electron k as an anti-symmetrized Slater determinant. However, at the same time the k (usually) forms a preponderant configuration, and it is an important fact67 that the relevant symmetry for the MO may not always be the point-group determined by the equilibrium nuclear positions but may be a higher symmetry. For many years, it was felt that the mathematical result (that a closed-shell Slater determinant contains k which can be arranged in fairly arbitrary new linear combinations by a unitary transformation without modifying k) removed the individual subsistence... 

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 appendix we generalise the expressions of the diabatic quantities first introduced in Sec. 2 for the ideal case of an exact two-level problem to a more realistic description. In a normal situation, the Hamiltonian has an infinite number of eigenstates, and there is no finite number of strictly diabatic states [76] that can describe a given pair of adiabatic states [77-80]. Instead, one can define a unitary transformation of the adiabatic states generating two quasidiabatic states characterised by a residual non-adiabatic coupling, as small as possible, but never zero (see, e.g., [5,24,32-35]). In practice, the electronic Hilbert space is always truncated to a finite number of configurations. In what follows, we consider the case of MCSCF wavefunctions and make use of generalised crude adiabatic states adapted to this. 

The two unitary operators evp[z/<(r)] and exp[ 5 (0] ensure that one is able to perform unitary transformations in the orbital and configuration space, respectively. 

Many text-books argue that penultimate orbitals should not be taken too seriously because a closed-shell anti-symmetrized Slater determinant is invariant when new linear combinations of the one-electron functions are formed by a unitary transformation. This mathematical truth is rather irrelevant for our purposes, in part because the actual total wave-function is not a well-defined configuration but shows correlation effects which can be ascribed to a first approximation (11, 24) to an internal dielectric screening of the interelectronic repulsion corresponding to admixture of configurations... 

The unitary transformation matrix A(q) of (71), (73) and (79) is arbitrary, and can be chosen to make (79) have desirable properties which (74) does not display. In many instances the couplings between consecutive PESs are significant only in the small regions of nuclear configuration space where those PESs are close to each other or... 

The branching multiplicity index is normally ctf > 1 since T, occurs more than once the equivalent states need to be distinguished by the index a. (Only in the special case of the d2 configuration and cubic symmetry does = 1.) The appropriate basis set kets transforming according to irreducible representation r, component y, of group G are then given by a unitary transformation... 

The unitary transformation of the reference state given in Eq. (2.18) has as generators the operators r. s of A and n><0 of S, It is possible that the operators r s and w><() span the same space. That is, the effects of the operators r. s may be expressed in terms of those of the stale projections in the configuration space. 1 o determine whether the elTects of a given operator can be expressed in terms of the kels m>, we examine the following difference ket ... 

The search for redundant variables may, of course, alternatively be performed in the configuration space 0 > since this space is related to the space /> through a unitary transformation. Because the states are normalized to unity, the search for redundant variables may be achieved by investigating whether the sum... 

The natural solution of the operator Eq. 4.38 would be in the momentum space due to the presence of momentum p which replaces the standard configuration space formulation by a Eourier transform. The success of the Douglas-Kroll-Hess and related approximations is mostly due to excellent idea of Bernd Hess [53,54] to replace the explicit Eourier transformation by some basis set (discrete momentum representation) where momentum p is diagonal. This is a crucial step since the unitary transformation U of the Dirac Hamiltonian can easily be accomplished within every quantum chemical basis set program, where the matrix representation of the nonrelativistic kinetic energy T = j2m is already available. Consequently, all DKH operator equations could be converted into their matrix formulation and they can be solved by standard algebraic techniques [13]. 

An alternative formulation of the MCSCF wavefunction makes use of an exponential unitary transformation of the orbitals and also of the configuration state functions in a given initial wavefunction... 

Configuration state functions (spin-adapted Slater determinants) constructed from excitation-adapted molecular orbitals (EAMOs) possess minimal off-diagonal elements of the Hamiltonian matrix. These orbitals, which result from separate unitary transformations among the occupied and virtual MOs. offer the most concise description of electronic excited states in terms of electron jumps . For example, at the CIS/6-31 +G level of theory, a symmetric combination of Just two singly excited configurations built from EAMOs [0.7049(7 — 9) -I-0.7049(6 -> 8)] suffices to adequately describe the first triplet excited state of N2, whereas several configurations involving MOs [0.5975 (7 9) -I- 0.5975(6 8) -t- 0.3646(7 16) -l-0.3646(6 15) -I- 0.0858(7 23) -I- 0.0858(6 22)J are... 

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