Multireference double excitation

For longer polyenes, multireference double excitation configuration interaction (MRD-CI) approximations have been employed which select the most important electronic configurations and balance the effects of electron correlation in the ground and excited states. The MRD-CI methods provide results comparable to Q-CI or full -Cl calculations on short polyenes (Zoos and Ramasesha, 1984) and have allowed calculations to be extended to molecules with up to eight double bonds (Tavan and Schulten,... 

QDMBPT = quasi-degenerate many-body perturbation theory, (POL-) Cl = (polarization-) configuration interaction, MC SCF = multiconfiguration SCF, CAS SCF = complete active space SCF, MRD Cl = multireference double-excitation Cl, UHF = unrestricted Hartree-Fock, UMP2 = unrestricted Moller-Plesset perturbation method of second order. - An analytical, combined polynomial-exponential form for E(x) with x = (r-rg)/r and a figure for x= —0.98 to +0.04. 

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)". 

The second step of the calculation involves the treatment of dynamic correlation effects, which can be approached by many-body perturbation theory (62) or configuration interaction (63). Multireference coupled-cluster techniques have been developed (64—66) but they are computationally far more demanding and still not established as standard methods. At this point, we will only focus on configuration interaction approaches. What is done in these approaches is to regard the entire zeroth-order wavefunc-tion Tj) or its constituent parts double excitations relative to these reference functions. This produces a set of excited CSFs ( Q) that are used as expansion space for the configuration interaction (Cl) procedure. The resulting wavefunction may be written as... 

MRCISD Multireference Cl including single and double excitations... 

In practice, one often has to restrict the active space to include only a few of the highest occupied and lowest unoccupied molecular orbitals. The correlation effects not included in the CASSCF calculation can then be recovered by a multireference Cl (MRCI) calculation, in which all single and double excitations from the CASSCF reference are taken into account [27], A computationally more efficient way of including dynamical electron correlation effects is perturbation theory with respect to the CASSCF reference. The most widely employed method of this type is the CASPT2 method developed by Roos and collaborators [28], The CASSCF and CASPT2 methods have been essential tools for the calculations described in this contribution. 

Modular FORTRAN programs for performing general ab initio multireference single- and double-excitation configuration interaction, averaged coupled-pair functional and linearized couple-cluster method calculations. Cray and other versions. 

It would be remiss not to mention multireference techniques for the high-level description of correlation within the coupled-cluster framework.2,3,26 Consider that double excitations from reference functions that are doubly excited relative to each other incorporate quadruple excitations (e.g., J4) from a single reference. However, further discussion of MR-CCM is beyond the scope of the present article, and we refer the interested reader to Refs. 2, 3, and 26. 

An alternative contraction scheme which has received more attention is internally contracted multireference CISD (usually denoted simply CMRCI), which was first discussed by Meyer99 and Siegbahn.100 This method applies the single and double excitation operators to a single multiconfigurational reference wavefunction as a whole, including the reference coefficients. Thus, if the reference wavefunction is... 

A simple way to implement n-particle space truncation is to use the uncorrelated wave function (which as noted above is a very substantial fraction of the exact wave function) to classify terms in the n-particle space. If we consider the Hartree-Fock determinant, for example, we can construct all CSFs in the full n-particle space by successively exciting one, two,.., electrons from the occupied Hartree-Fock MOs to unoccupied MOs. For cases in which a multiconfigurational zeroth-order wave function is required, the same formal classification can be applied. Since only singly and doubly excited CSFs can interact with the zeroth-order wave func tion via the Hamiltonian in Eq. (1), it is natural to truncate the n-particle expansion at this level, at least as a first approximation. We thus obtain single and double excitations from Hartree-Fock (denoted SDCI) or its multiconfigurational reference analog, multireference Cl (MRCI). 

The classification of CSFs into single, double,... excitations is straightforward and unambiguous for a closed-shell Hartree-Fock reference function. In open-shell or multireference cases, there are more possibilities for defining these excited CSFs, some of which interact with the reference space in the lowest order of perturbation theory, and some of which do not. It is very common to exclude the latter excitations. This is termed restricting the wave function to the first-order interacting space. ... 

---