Reference space

All of the CSFs in the SCF (in which case only a single CSF is included) or MCSCF wavefunction that was used to generate the molecular orbitals (jii. This set of CSFs are referred to as spanning the reference space of the subsequent CI calculation, and the particular combination of these CSFs used in this orbital optimization (i.e., the SCF or MCSCF wavefunction) is called the reference function. 

The notation for denoting this type of calculation is sometimes more specihc. For example, the acronym MCSCF+1+2 means that the calculation is a MRCI calculation with single and double Cl excitations out of an MCSCF reference space. Likewise, CASSCF+1+2 and GVB+1+2 calculations are possible. 

CIS calculations from the semiempirical wave function can be used for computing electronic excited states. Some software packages allow Cl calculations other than CIS to be performed from the semiempirical reference space. This is a good technique for modeling compounds that are not described properly by a single-determinant wave function (see Chapter 26). Semiempirical Cl... 

Intensities for electronic transitions are computed as transition dipole moments between states. This is most accurate if the states are orthogonal. Some of the best results are obtained from the CIS, MCSCF, and ZINDO methods. The CASPT2 method can be very accurate, but it often requires some manual manipulation in order to obtain the correct configurations in the reference space. 

The present approach is one of the second-generation multireference perturbation treatments first opened by the CIPSI algorithm 20 years ago. Even if the spirit of these new treatments is different, mainly because the reference space is chosen on its completeness rather than on energetical criteria, it remains that the unavoidable problems of disk storage, bottleneck of variational approaches, can now be conveniently transferred to the problem of CPU time which is less restrictive. 

All three states were described by a single set of SCF molecular orbitals based on the occupied canonical orbitals of the X Z- state and a transformation of the canonical virtual space known as "K-orbitals" [10] which, among other properties, approximate the set of natural orbitals. Transition moments within orthogonal basis functions are easier to derive. For the X state the composition of the reference space was obtained by performing two Hartree-Fock single and double excitations (HFSD-CI) calculations at two typical intemuclear distances, i.e. R. (equilibrium geometry) and about 3Re,and adding to the HF... 

An avoided crossing did occur at 3R which, however, was absent at 5.53ao [8].Thus the combined space spanned by all configurations whose coefficients equalled or exceeded 0.03 in either of the MRSD-Cls (at Re and 5.53Re) was chosen for the reference space to... 

Estimates of the energy contributions from higher than double excitations out of the reference space were obtained by means of one form of the "Davidson correction" [11,7]. More details can be found in references [7,8]. 

These single reference-based methods are limited to cases where the reference function can be written as a single determinant. This is most often not the case and it is then necessary to use a multiconfigurational approach. Multrreference Cl can possibly be used, but this method is only approximately size extensive, which may lead to large errors unless an extended reference space is used. For example, Osanai et al. [8] obtained for the excitation energy in Mn 2.24 eV with the QCISD(T) method while SDCI with cluster corrections gave 2.64 eV. Extended basis sets were used. The experimental value is 2.15 eV. 

Protein Synthetic oligonucleotide (5 -3 ) Res (A) References Space group P DNA ratio T C Buffer and additives... 

Since the definition of chemical reference spaces very much depends on the choice of molecular descriptors, we begin the description with a brief overview of some commonly used types of descriptors, as summarized in Table 1. 

Once -dimensional chemical reference space has been defined, the descriptors values are calculated for all compounds in a dataset, thereby assigning a coordinate vector to each molecule. In principle, partitioning analysis could proceed in -dimensional space, but it is often attempted to reduce its dimensionality in order to generate a low-dimensional representa-... 

In contrast to partitioning methods that involve dimension reduction of chemical reference spaces, MP is best understood as a direct space method. However, -dimensional descriptor space is simplified here by transforming property descriptors with continuous or discrete value ranges into a binary classification scheme. Essentially, this binary space transformation assigns less complex -dimensional vectors to test molecules, with each dimension having unity length of either 0 or 1. Thus, although MP analysis proceeds in -dimensional descriptor space, its dimensions are scaled and its complexity is reduced. 

A major practical issue affecting MP calculations is caused by use of correlated molecular descriptors. During subsequent MP steps, exact halves of values (and molecules) are only generated if the chosen descriptors are uncorrelated (orthogonal), as shown in Fig. 1A. By contrast, the presence of descriptor correlations (and departure from orthogonal reference space) leads to overpopulated and underpopulated, or even empty, partitions (see also Note 5), as illustrated in Fig. ID. For diversity analysis, compounds should be widely distributed over computed partitions and descriptor correlation effects should therefore be limited as much as possible. However, for other applications, the use of correlated descriptors that produce skewed compound distributions may not be problematic or even favorable (see Note 5). 

The main reason why existing MR CC methods as well as related MR MBPT cannot be considered as standard or routine methods is the fact that both theories suffer from the Intruder state problem or generally from the convergence problems. As is well known, both MR MBPT/CC theories are built on the concept of the effective Hamiltonian that acts in a relatively small model or reference space and provides us with energies of several states at the same time by diagonalization of the effective Hamiltonian. In order to warrant size-extensivity, both theories employ the complete model space formulations. Although conceptually simpler, the use of the complete model space makes the calculations rather... 

In general, we do not need to know the whole energy spectrum, but we are interested in several low lying states, say, where a = 1,2,...,d. Let us further assume that the most important contributions to d exact wave functions iP, come from d configurations represented by Slater determinants in the spin-orbital formalism, where ft = 1,2,..., d. Given dominant configurations span the so-called model or reference space. To simplify... 

It turns out that MR CISD represents again the most suitable source of the required higher-order clusters. Carefully chosen small reference space MR CISD involves a very small, yet representative, subset of such cluster amplitudes. Moreover, in this way we can also overcome the eventual intruder state problems by including such states in MR CISD, while excluding them from CMS SU CCSD. In other words, while we may have to exclude some references from Ado in order to avoid intruders, we can safely include them in the MR CISD model space Adi. In fact, we can even choose the CMS for Adi. Thus, designating the dimensions of Ado and Adi spaces by M and N, respectively, we refer to the ec SU CCSD method employing an NR-CISD as the external source by the acronym N, M)-CCSD. Thus, with this notation, we have that (N, 1)-CCSD = NR-RMR CCSD and (0, M)-CCSD = MR SU CCSD. Also, (0,1)-CCSD = SR CCSD. For details of this procedure and its applications we refer the reader to Refs. [63,64,71]. 

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