MRPT

SF-OD level with the basis set composed of a cc-pVTZ basis on carbons and a cc-pVDZ basis on hydrogens). These energies are very close to the MRPT values (26) of 0.72 and 0.83 eV (for the 1 fi and 1 Ai states, respectively). With regard to experiment, the lowest adiabatic state, 1 B, has not been observed in the photoelectron spectrum (40) because of unfavorable Frank-Condon factors. The experimental adiabatic energy gap (including ZPE) between the ground triplet state and the VA state is 0.70 eV. The estimated experimental >s 0.79 eV, which is 0.15 eV lower than the SF-OD estimate. 

Figure 1. The shape of the potential curve for nitrogen in a correlation-consistent polarized double-zeta basis set is presented for the variational 2-RDM method as well as (a) single-reference coupled cluster, (b) multireference second-order perturbation theory (MRPT) and single-double configuration interaction (MRCl), and full configuration interaction (FCl) wavefunction methods. The symbol 2-RDM indicates that the potential curve was shifted by the difference between the 2-RDM and CCSD(T) energies at equilibrium.

Equilibrium Bond Distance and the Harmonic Frequency for N2 from the 2-RDM Method with 2-Positivity (DQG) Conditions Compared with Their Values from Coupled-Cluster Singles-Doubles with Perturbative Triples (CCD(T)), Multireference Second-Order Perturbation Theory (MRPT), Multireference Configuration Interaction with Single-Double Excitations (MRCI), and Full Configuration Interaction (FCI)". 

Density functional theory has also been applied to the Cope rearrangement. Nonlocal methods, such as BLYP and B3LYP, find a single transition state with approximately 2 A. The barrier height is in excellent agreement with experiment. These first DFT results were extremely encouraging because DFT computations are considerably less resonrce-intensive than MRPT. Moreover, analytical first and second derivatives are available for DFT, allowing for efficient optimization of stmc-tures (particularly transition states) and the computation of vibrational frequencies needed to characterize the nature of the stationary points. Analytical derivatives are not available for MRPT calculations, which means that there is a more difficult optimization procedure and the inability to fully characterize structures. 

When MCSCF wavefunctions are used as the reference, the most commonly used methods for recovering the electron correlation are multireference configuration interaction (MRCI) (15) and multireference perturbation theory (MRPT)... 

The former is usually implemented at the single- and double-excitation level, MR(SD)CI, but it is so computationally demanding that this level of theory is still limited to small active spaces. Second-order MRPT is more efficient, as indeed is its single-reference analog. 

Although not rigorously size-extensive, CASPT behaves better than MRSDCI theory for systems of many electrons. The CASPT and other MRPT methods are important in the sense that they are the only generally applicable methods for ab initio calculations of dynamical correlation of open-shell and closed-shell multiconfigura-... 

An alternative way to approach the problem is to start out with a hxed MR function, and develop a perturbation theory on it. This is thus an MRPT of the unrelaxed or contracted coefficients variety. The SSMR-based perturbation theories based on contracted description using CAS [43-48] have been widely used as efficient methods to treat quasi-degeneracy. There are usually two ways in which the virtual functions are handled they can be contracted functions themselves or they can be simpler CSFs. Multistate versions of the contracted variety has also been suggested [46]. An SS-based CC formulation of the. frozen variety has been developed [38], where a wave operator is used to generate the exact state by its action on the entire MR function. 

A second order MRPT for the restricted active space (RAS) and a general active space was proposed [63]. More recently another theory has been developed [64] using a QCAS (Quasi-Complete Active Space). These theories, unlike in Ref. [60], are not rigorously size-extensive. 

Another approach for treating the quasi-degeneracy is adopted by the various MR-based CEPA methods, which have appeared parallely along with the MRCC and MRPT methods. The earlier developed state-specific MRCEPA methods [37,65-70] avoided the redundancy problem using non-redundant cluster operators to compute the dynamical correlation on the zeroth order MR wave function. The MR version of (SC) CI method, termed as MR-(SC) CI [37], can be viewed as the size-extensive dressing of the MR-CISD method just as the (SC) CI [71] is considered to be the size-extensive dressing of the SR-CISD method. Similar to the SR-case, they include all EPV terms in an exact manner. 

This article is organized as follows in Section 22.2.1 we will first describe the formulation of the spin-free SS-MRCC theory. We wUl next present the development of the API-SSMRCC method in Section 22.2.2. In Section 22.2.3 we will discuss the aspects of size-extensivity and consistency of the API-SSMRCC method. We will discuss the spin-free formulation of SS-MRPT and SS-MRCEPA methods starting from the SS-MRCC theory in Sections 22.3 and 22.4, respectively. The IMS version of the SS-MRCC method will be covered in Section 22.5. In Section 22.6, we will present the illustrative numerical applications, along with discussions. Einally, Section 22.7 will summarize our presentation. 

EMERGENCE OE STATE-SPECIFIC MULTI-REFERENCE PERTURBATION THEORY SS-MRPT FROM SS-MRCC THEORY... 

Eqs. (36) and (37) are our principal working equations for RS-based SS-MRPT. It is noteworthy that in the RS-based SS-MRPT formalism the zeroth order coefficients, c s are used to evaluate the cluster operators in Eq. (36), but the coefficients are relaxed during the computation of E since this is obtained by diagonalization via Eq. (37). 

The robustness of the energy denominators in the presence of intruders is quite manifest in our SS-MRPT formalism. The denominator in Eq. (36) is never small as long as the unperturbed or the perturbed energy, Eq, is well separated from the energies of the virtual functions. Thus the SS-MRPT is intruder-free, and explicitly size-extensive and also size-consistent when we use orbitals localized on the separated fragments. 

Although the approach described above is presented in its most general form, using a multiple coupled-cluster Ansatz for the SS-MRCC formalism, suitable approxi-mants to it such as the state-specific multi-reference perturbation theory (SS-MRPT) or state-specific multi-reference CEPA (SS-MRCEPA) can be generated by straightforward approximations. Since the new closed component of the wave operator for IMS appear first at the quadratic power, it is evident that the expressions we have derived in this and the earlier papers for the CAS will remain valid if the quadratic powers of are ignored in the approximants to SS-MRCC for IMS. This implies that all the SS-MRPT... 

The results with the SS-MRCEPA(O) and SS-MRCEPA(I) were computed by us earlier in another paper [59]. Here we summarize the deviation of the SS-MRCEPA results with respect to the FCI values reported earlier [59] to compare the relative performance of our SS-MRPT methods. The minimum and maximum deviations of SS-MRCEPA(I) method from FCI are approximately of the order of 0.45 and 0.7 mH, respectively, along with the average deviation being 0.6 mH whereas the corresponding values for the deviations in case of CEPA(O) are 0.1 and 0.5 mH, respectively. The average deviation for the CEPA(O) method is of the order of 0.4 mH. Hence we may conclude that the relative performance of our SS-MRPT(EN) is pretty close to the SS-MRCEPA(I) method for the H4 model system. 

Since the two active orbitals belong to different symmetries, the CAS is two-dimensional and both the model functions are closed. The SS-MRCC theory is trivially spin-adapted in this case, and the performance of the various SS-MRPT variants can be assessed with respect to both the FCI results and the SS-MRCC results. 

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