First Unitary Transformation

In order to eliminate the odd operator Oq of the Dirac Hamiltonian as written in Eq. (11.40) order by order in the scalar potential Y, an odd operator that depends on V must be generated. As discussed in section 11.5, only the special closed-form free-particle Foldy-Wouthuysen transformation produces an operator 0 linear in V as indicated by the subscript. This is the mandatory starting point for subsequent transformation steps. 

For convenience we recall the result of the closed-form free-particle Foldy-Wouthuysen transformation of Eq. (11.35)  

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Incidentally if t — t0 is an infinitesimal time dt, the unitary transformation reduces in first order to... 

The existence of a unitary transformation U(a,A) which relates the field operators in the two frames imposes certain conditions on the operators themselves they must satisfy certain commutation rules with P and Muv. Consider first the case of a space-time translation... 

Let us briefly mention some formal aspects of the above-introduced formalism, which have been discussed in detail by Blaizot and Marshalek [218]. First, it is noted that the both the Schwinger and the Holstein-Primakoff representations are not unitary transformations in the usual sense. Nevertheless, a transformation may be defined in terms of a formal mapping operator acting in the fermionic-bosonic product Hilbert space. Furthermore, the interrelation of the Schwinger representation and the Holstein-Primakoff representation has been investigated in the context of quantization of time-dependent self-consistent fields. It has been shown that the representations are related to each other by a nonunitary transformation. This lack of unitarity is a consequence of the nonexistence of a unitary polar decomposition of the creation and annihilation operators a and at [221] and the resulting difficulties in the definition of a proper phase operator in quantum optics [222]. 

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. 

We first consider a Hamiltonian, thus deterministic, system. Denoting by oo the set of all phase space coordinates of a point in phase space (which determines the instantaneous state of the system), the motion of this point is determined by a canonical transformation evolving in time, 7), with Tq = I. The function of time TfCO thus represents the trajectory passing through co at time zero. The evolution of the distribution function is obtained by the action on p of a unitary transformation Ut, related to 7) as follows ... 

The localized many-body perturbation theory (LMBPT) applies localized HF orbitals which are unitary transforms of the canonical ones in the diagrammatic many-body perturbation theory. The method was elaborated on models of cyclic polyenes in the Pariser-Parr-Pople (PPP) approximation. These systems are considered as not well localized so they are suitable to study the importance of non local effects. The description of LMBPT follows the main points as it was first published in 1984 (Kapuy etal, 1983). 

In this manner, we have arrived at the Pernal nonlocal potential [81]. It can be shown, using the invariance of Vee with respect to an arbitrary unitary transformation and its extremal properties [13] or by means of the first-order perturbation theory applied to the eigenequation of the 1-RDM [81], that the off-diagonal elements of Uee may also be derived via the functional derivative... 

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... 

It is worth noting that the methods mentioned give rise to Hartree-Fock equations based on different Fock operators for the various orbitals of the same spin. This implies that off-diagonal Lagrange multipliers appear which cannot be eliminated by a suitable unitary transformation. Moreover, the complications arise when excited states above the first one are considered (see, e.g. [25] for further details). 

In the language of control theory, Tr[p(0)P] is a kinematic critical point [87] if Eq. (4.159) holds, since Tr[e p(0)e- P] = Tr[p(0)P] + Tr(7/[p(0),P]) + O(H ) for a small arbitrary system Hamiltonian H. Since we consider p in the interaction picture, Eq. (4.159) means that the score is insensitive (in first order) to a bath-induced unitary evolution (i.e., a generalized Lamb shift) [88]. The purpose of this assumption is only to simplify the expressions, but it is not essential. Physically, one may think of a fast auxiliary unitary transformation that is applied initially in order to diagonalize the initial state in the eigenbasis of P. 

To construct the Fock matrix, one must already know the molecular orbitals ( ) since the electron repulsion integrals require them. For this reason, the Fock equation (A.47) must be solved iteratively. One makes an initial guess at the molecular orbitals and uses this guess to construct an approximate Fock matrix. Solution of the Fock equations will produce a set of MOs from which a better Fock matrix can be constructed. After repeating this operation a number of times, if everything goes well, a point will be reached where the MOs obtained from solution of the Fock equations are the same as were obtained from the previous cycle and used to make up the Fock matrix. When this point is reached, one is said to have reached self-consistency or to have reached a self-consistent field (SCF). In practice, solution of the Fock equations proceeds as follows. First transform the basis set / into an orthonormal set 2 by means of a unitary transformation (a rotation in n dimensions),... 

The functions 7, Eq. 4.8, were obtained by first coupling the spherical harmonics in the variables Qi and Q2, and subsequently coupling the result with the spherical harmonics in 2. This particular coupling scheme we label (QiQ2 Q), it leads to a particular set of coefficients A in Eq. 4.7 which is our standard set. Obviously, two other coupling schemes are possible, namely (Q2Q fli) and (flifl fl2). These lead to sets of coefficients A and A" which are related to the set A by certain unitary transforms discussed elsewhere [323, 391]. 

First, let the unitary transformation diagonalizes the interaction matrix that involves the operators of the electron repulsion, the crystal field, and the... 

To obtain the matrix K= [ 1, we first make a unitary transformation,... 

The first reason that led Latora and Baranger to evaluate the time evolution of the Gibbs entropy by means of a bunch of trajectories moving in a phase space divided into many small cells is the following In the Hamiltonian case the density equation must obey the Liouville theorem, namely it is a unitary transformation, which maintains the Gibbs entropy constant. However, this difficulty can be bypassed without abandoning the density picture. In line with the advocates of decoherence theory, we modify the density equation in such a way as to mimic the influence of external, extremely weak fluctuations [141]. It has to be pointed out that from this point of view, there is no essential difference with the case where these fluctuations correspond to a modified form of quantum mechanics [115]. 

The appearance of one or more CT-excitons below the conduction band of the conjugated chain may not appear to be of major importance for the properties of the material. In fact the consequences of the occurrence of excitons are significant. Slater and Shockley (1936) demonstrated that the descriptions of the system by Bloch functions, i.e. the band model, and by localised excitations, i.e. excitons, were related to one another by a unitary transformation. They were also the first to consider the impact of the... 

In quantum mechanics the unitary transformation (8) leads to different gauges for the Hamiltonian, which are sometimes also referred to as different formalisms. An alternative first-order Hamiltonian can be defined from (8)... 

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. 

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