Doubly excited function

The last equality follows from the fact that each term is equal to zero. The first vanishes since both determinants differ by four excitations. Indeed, (J denotes a double deexcitation of the doubly excited function (i.e., something proportional to (ol)- For similar reasons (too strong deexcitations give zero), the remaining terms in A also vanish. As a result, we need to solve the equation... 

When a CASSCF wave function is used for a MRCISD calculation, the number of CSFs produced may be too many to deal with, so various methods are used to reduce the amount of computation needed. One widely used procedure is internally contracted MRCI (icMRCI) [H.-J. Werner and P. J. Knowles, J. Chem. Phys., 89,5803 (1988)]. Here, the optimized MCSCF function is treated as a single reference function (with fixed coefficients) from which one generates doubly excited functions. Each excited function is a linear combination of many ordinary CSFs, with the coefficients within a given excited function being held fixed at the values for the MCSCF function. Thus, far fewer coefficients need to be calculated than in a conventional (uncontracted) MRCI calculation. (Singly excited functions are also included, but for technical reasons, these are not contracted but are treated as in uncontracted MRCI.) Experience has shown that the contracted MRCI wave function is almost as accurate as the uncontracted one. 

The doubly excited function is an eigenfunction of with an eigen-... 

From the Brillouin theorem, the Cl involves only [Pg.316]

Let us then consider the case where the degenerate mode is doubly excited. In this case, V2 = V2a + V2i = 2 and the coiTesponding vibrational energy level will be triply degenerate with the associated wave functions being given by... 

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... 

The amount of computation for MP2 is determined by the partial transformation of the two-electron integrals, what can be done in a time proportionally to m (m is the number of basis functions), which is comparable to computations involved in one step of CID (doubly-excited configuration interaction) calculation. To save some computer time and space, the core orbitals are frequently omitted from MP calculations. For more details on perturbation theory please see A. Szabo and N. Ostlund, Modem Quantum Chemistry, Macmillan, New York, 1985. 

The relative importance of tlie different excitations may qualitatively be understood by noting tliat the doubles provide electron correlation for electron pairs, Quadruply excited determinants are important as they primarily correspond to products of double excitations. The singly excited determinants allow inclusion of multi-reference charactei in the wave function, i.e. they allow the orbitals to relax . Although the HF orbitals are optimum for the single determinant wave function, that is no longer the case when man) determinants are included. The triply excited determinants are doubly excited relative tc the singles, and can then be viewed as providing correlation for the multi-reference part of the Cl wave function. 

The optimum values of die oq and a coefficients are determined by the variational procedure. The HF wave function constrains both electrons to move in the same bonding orbital. By allowing the doubly excited state to enter the wave function, the electrons can better avoid each other, as the antibonding MO now is also available. The antibonding MO has a nodal plane (where opposite sides of this plane. This left-right correlation is a molecular equivalent of the atomic radial correlation discussed in Section 5.2. 

The dissociation problem is solved in the case of a full Cl wave function. As seen from eq. (4.19), the ionic term can be made to disappear by setting ai = —no- The full Cl wave function generates the lowest possible energy (within the limitations of the chosen basis set) at all distances, with the optimum weights of the HF and doubly excited determinants determined by the variational principle. In the general case of a polyatomic molecule and a large basis set, correct dissociation of all bonds can be achieved if the Cl wave function contains all determinants generated by a full Cl in the valence orbital space. The latter corresponds to a full Cl if a minimum basis is employed, but is much smaller than a full Cl if an extended basis is used. 

The formula for the first-order correction to the wave function (eq. (4.37)) similarly only contains contributions from doubly excited determinants. Since knowledge of the first-order wave function allows calculation of the energy up to third order (In - - 1 = 3, eq. (4.34)), it is immediately clear that the third-order energy also only contains contributions from doubly excited determinants. Qualitatively speaking, the MP2 contribution describes the correlation between pairs of electrons while MP3 describes the interaction between pairs. The formula for calculating this contribution is somewhat... 

In the MO-CI language, the correct dissociation of a single bond requires addition of a second doubly excited determinant to the wave function. The VB-CF wave function, on the other hand, dissociates smoothly to the correct limit, the VB orbitals simply reverting to their pure atomic shapes, and the overlap disappearing. 

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... 

It is instructive to discuss the MP2-R12 method [37] before going into more involved CC-R12. As in MP2, the wave function of MP2-R12 (IT1)) is a linear combination of the reference HF determinant ( o)) and doubly excited determinants produced by the action of a two-electron excitation operator T ... 

Equation (2.18) is a linear variation function. (The summation indices prevent double-counting of excited configurations.) The expansion coefficients cq, c, c%, and so on are varied to minimize the variational integral. o) is a better approximation than l o)- In principle, if the basis were complete. Cl would provide an exact solution. Here we use a truncated expansion retaining only determinants D that differ from I Tq) by at most two spin orbitals this is a singly-doubly excited Cl (SDCI). 

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