Reference Functions

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 first-order MPPT wavefunction can be evaluated in terms of Slater determinants that are excited relative to the SCF reference function k. Realizing again that the perturbation coupling matrix elements I>k H i> are non-zero only for doubly excited CSF s, and denoting such doubly excited i by a,b m,n the first-order... 

There are two reference functions IT (T q), one for the range from 13 K to 0.01°C, another for the range 0.01—962°C. The reference functions represent the caUbration of a fictitious thermometer developed from experience in the caUbration of many SPRTs over many years. Below 0.01°C, the reference function is... 

Table A2.3 Values of the coefficients A,-, B, Cand D, and of the constants A0, Bo, Co, and Do in the reference functions in equations (A2.6) and (A2.7) and in the inverse functions given by equations (A2.8) and (A2.9). These functions, coefficients, and constants are part of the definition of ITS-90.

Using the above partitioning into the Rayleigh-Schrddinger perturbation theory (RSPT) allows the perturbed reference function to be written as,... 

Here, /j and rj are the l" left- and the J right-hand eigenvectors of the non-Hermitian Hamiltonian H. The operator is represented on the space spanned by the manifold created by the excitations out of a Hartree-Fock reference determinant, including the null excitation (the reference function). When we calculate the transition probability between a ground state g) and an excited state ]e), we need to evaluate and The reference function is a right-... 

Andersson K, Malmqvist PA, Roos BO, Sadlej AJ, Wolinski K (1990) Second-order perturbation-theory with a CASSCF reference function. J Phys Chem 94 5483... 

Nakano H (1993) Quasidegenerate perturbation theory with multiconfigurational self-consistent-field reference functions. J Chem Phys 99 7983... 

Multiconfigurational Self-Consistent-Field Reference Functions. 

Equation [1] is an internally contracted configuration space, doubly excited with respect to the CAS reference function 0) = G4SSCF) one or two of the four indices p,q,r,s must be outside the active space. The functions of Eq. [1] are linear combinations of CFs and span the entire configuration space that interacts with the reference function. Labeling the compound index pqrs as (i or v, we can write the first-order equation as... 

Here, Flffl are matrix elements of a zeroth-order Hamiltonian, which is chosen as a one-electron operator in the spirit of MP2. is an overlap matrix The excited CFs are not in general orthogonal to each other. Finally, Vf)(i represents the interaction between the excited function and the CAS reference function. The difference between Eq. [2] and ordinary MP2 is the more complicated structure of the matrix elements of the zeroth-order Hamiltonian in MP2 it is a simple sum of orbital energies. Here H is a complex expression involving matrix elements of a generalized Fock operator F combined with up to fourth-order density matrices of the CAS wave function. Additional details are given in the original papers by Andersson and coworkers.17 18 We here mention only the basic principles. The zeroth-order Hamiltonian is written as a sum of projections of F onto the reference function 0)... 

The reference (zeroth-order) function in the CASPT2 method is a predetermined CASSCF wave function. The coefficients in the CAS function are thus fixed and are not affected by the perturbation operator. This choice of the reference function often works well when the other solutions to the CAS Hamiltonian are well separated in energy, but there may be a problem when two or more electronic states of the same symmetry are close in energy. Such situations are common for excited states. One can then expect the dynamic correlation to also affect the reference function. This problem can be handled by extending the perturbation treatment to include electronic states that are close in energy. This extension, called the Multi-State CASPT2 method, has been implemented by Finley and coworkers.24 We will briefly summarize the main aspects of the Multi-State CASPT2 method. 

Second-Order Perturbation Theory with a CASSCF Reference Function. 

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. 

The optimization of excited states of type (1) or (3) is usually straightforward, but excited states of type (2) may require care in the choice of reference function [9]. 

While particle contractions connect an upper right with a lower left label (and are associated with a factor (1 — np)), hole contractions go from upper left to lower right, with a factor —np. Closed loops introduce another factor —LA graphical interpretation is possible, in agreement with that for the conventional particle-hole picture. We postpone this to the more general case of an arbitrary reference function (Section III.E). 

As in the previous section we consider a single Slater determinant reference function with the spin orbitals i/, occupied. However, we express our excitation operators in a completely arbitrary basis of spin orbitals i/, which is no longer the direct sum of occupied and unoccupied spin orbitals. Then the following replacements must be made [3] ... 

C. Normal Ordering with Respect to Arbitrary Reference Function... 

In order to generalize the concept of normal ordering such that it is valid with respect to any arbitrary reference function F, we start from the following guiding principles ... 

For a more general reference function there are not only particle and hole contractions, but also contractions that involve cumulants. Again it holds that the expectation value of a product of d operators with respect to the reference function T is equal to the sum of all full contractions ... 

Note that a dot ( ) always means a matrix element of the antisymmetrized electron interaction g, a cross (x) a matrix element of the one-particle operator /, while an open square ( ) collects the free labels in any of these contractions. If the reference function is a single Slater determinant, all cumulants X vanish one is then left with particle and hole contractions, like in traditional MBPT in the particle-hole picture. 

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