Hilbert space truncated states

First, as the molecule on which the chromophore sits rotates, this projection will change. Second, the magnitude of the transition dipole may depend on bath coordinates, which in analogy with gas-phase spectroscopy is called a non-Condon effect For water, as we will see, this latter dependence is very important [13, 14]. In principle there are off-diagonal terms in the Hamiltonian in this truncated two-state Hilbert space, which depend on the bath coordinates and which lead to vibrational energy relaxation [4]. In practice it is usually too difficult to treat both the spectral diffusion and vibrational relaxation problems at the same time, and so one usually adds the effects of this relaxation phenomenologically, and the lifetime 7j can either be calculated separately or determined from experiment. Within this approach the line shape can be written as [92 94]... 

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. 

Figure 3. Generalized coherent states (black bars) versus truncated coherent states (white bars) photon-number distribution Ps(n) as a function of n in FD Hilbert spaces with s = 5,..., 50 for the same displacement parameters a = a = 4.

Analogously to the generalized, CS in a FD Hilbert space, analyzed in Section IV. A, other states of the electromagnetic field can be defined by the action of the FD displacement or squeeze operators. In particular, FD displaced phase states and coherent phase states were discussed by Gangopadhyay [28]. Generalized displaced number states and Schrodinger cats were analyzed in Ref. 21 and generalized squeezed vacuum was studied in Ref. 34. A different approach to construction of FD states can be based on truncation of the Fock expansion of the well-known ID harmonic oscillator states. The same construction, as for the... 

The generalized phase CS, p,0o)(s), and truncated phase CS, j3,0o)(s), are associated with the Pegg-Bamett formalism of the Hermitian phase operator S. The operators 4>s, Hilbert space Thus the generalized and truncated phase CS are properly defined only in of finite dimension. States p,0o)( and p, 0o)(s), similar to a)(s) and a)(s), approach each other for p 2 = p 2 < C s/n [20]. This can be shown explicitly by calculating the scalar product between generalized and truncated phase CS. We find (p = p)... 

In this approach, one begins by subdividing the total system into several blocks An and proceeds to iteratively build effective blocks so that at each iteration, each effective block represents two or more blocks of the previous iteration, without increasing the Fock space dimensionality of the blocks from what existed at the previous iteration. Usually, one starts with each An consisting of a single site. Since the Hilbert space grows exponentially with the increase in system size, one truncates the number of states kept at each iteration. The quantum RG procedure proceeds as follows ... 

Because one is usually only interested in solving for low-lying states, one first divides the Hilbert space spanned by Wp into two parts, using the projection operators P and Q. The truncated (or shell-model) space is defined by P, while Q defines the space outside the shell-model space consequently, P p = <>p in Eq. (2). It is assumed that the P and Q spaces are non-overlapping, i.e., PQ = 0. 

So far, we focused on conventional quantum chemical approaches that approximate the FCI wave function by truncating the complete N-particle Hilbert space based on predefined configuration selection procedures. In a different approach, the number of independent Cf coefficients can be reduced without pruning the FCI space. This is equivalent to seeking a more efficient parameterization of the wave function expansion, where the Cl coeflBcients are approximated by a smaller set of variational parameters that allow for an optimal representation of the quantum state of interest. Different approaches, which we will call modern solely to distinguish them from the standard quantum chemical methods, have emerged from solid-state physics. 

We wish to truncate and rotate the Hilbert space of the system block subject to retaining an optimal representation of ). Denoting the size of the truncated superblock Hilbert space as Mg and the approximate, optimal superblock state as 1 ) we have,... 

The procedure for constructing an optimally truncated Hilbert space of a single block (usually block 1 or 2) is a straightforward generalization of the DMRG method. Suppose that the Hilbert space of block j is to be tnmcated. Then we construct an environment block state from the remaining block states. 

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