Double excitation operators

At each excitation level beyond the single-excitation level, a number of terms contribute. For example, double excitations are generated both by means of the double-excitation operator T2 (connected excitations)... 

Of course, operating on the HF wave function with T is, in essence, full Cl (more accurately, in full Cl one applies 1 + T), so one may legitimately ask what advantage is afforded by the use of the exponential of T in Eq. (7.49). The answer lies in the consequences associated with truncation of T. For instance, let us say that we only want to consider the double excitation operator, i.e., we make the approximation T = T2. In that case, Taylor expansion of the exponential function in Eq. (7.49) gives... 

The operators Tn are n-electron excitation operators that excite n electrons from the occupied orbitals to the unoccupied orbitals. Alternatively, they may be said to replace n occupied spin orbitals in 0 > hv n unoccupied spin orbitals. Tn sums over all possible n-electron excitations combinations, and each excitation has its own weight that must be solved for (see below). The form of the single- and double-excitation operators is... 

The next step to recover the fifth-order energy diagrams is to operate with on the T(3) amplitude in order to obtain T. Since at the third-order level we have a single excitation operator and at the fourth-order level we need a double excitation operator, we have to use the form of the Vj operator that increases the level of excitation, i.e., -—V- When com-... 

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... 

We find that it is convenient to work with the spin-adapted form of the coupled-cluster doubles (CCD) equations. The spin-adapted double excitation operators S (i),i = 1,2, are given, for example, in Oddershede et al. (1984, Appendix C). 

The FCI state (Eq. (43)) has been generated from the Hartree-Fock state (Eq. (42)) by application of the double excitation operator as in Eq. (41), followed by a variational optimization. Whereas the one-electron density p( 7) represents the overall probability of finding an electron at a given point 7 in space, the two-electron density p(7, 77) represents the probability of finding one electron at position 7 when the other electron is known to be at 7z-... 

One of the special cases of coupled-cluster theory is the singles-and-doubles (CCSD) model [37]. The cluster operator Eq. (29) is restricted to contain only the singles and doubles excitation operators. The importance of this model can be seen from the fact that, for any coupled-cluster wave function, the singles and doubles amplitudes are the only ones that contribute directly to the coupled-cluster energy. In the explicitly correlated CCSD model the conventional cluster operator containing the T and T2 operators is supplemented with an additional term that takes care of the explicit correlation (written with red font)... 

Contracted versus uncontracted The simplest way to define the first-order wave function is to apply single and double excitation operators on all the determinants (or CSFs) of the reference wave function. 

A break through for the application of the CC methods to excited states of large systems has recently been obtained by Christiansen et al. These authors introduced a simplified LR-CCSD method termed CC2, where the doubles excitation operator is taken from second-order perturbation theory (Qi)- The CC2 ground-state wave function can be written as... 

In addition to these second-order corrections to the RPA matrices there are three new matrices due to the operators. In the following, we present explicit expressions for them in terms of spatial orbitals (f>p and for two spin-free operators Pa and using a biorthogonal set of double excitation operators qq (Bak et ai, 2000). 

Coupled cluster with singles and doubles excitations (CCSD) is a size-consistent post-HF electron correlation method. The wavefunction, Y, in coupled cluster theory is formulated in terms of a cluster (exponential) expansion including the single and double excitation operators 7i and %. The effect of triple excitations (T) is calculated with perturbation theory. 

Ti being a single excitation operator, T2 a double excitation operator and so on up to Tn for an n-electron excitation. 

Commutation occurs since the excitation operators always excite from the set of occupied Hartree-Fock spin orbitals to the virtual ones - see (13.1.2) for the double-excitation operators. The creation and annihilation operators of the excitation operators therefore anticommute. 

From (13.1.11) it is apparent that a coupled-cluster wave function - generated, for example, by all possible single- and double-excitation operators - contains contributions from all determinants entering the FCl wave function (although the number of free parameters is usually much smaller). In practice, therefore, we cannot work with the coupled-cluster state in the expanded form (13.1.11) but we must instead retain the wave function in the more compact form (13.1.7), avoiding references to the individual determinants. 

Thus, in addition to the density-matrix elements for lla and lcr ) determined in Exercise 5.2.1, we must consider the transition-density elements. The one-electron transition-density elements in (5S.2.7) vanish since a single-excitation operator cannot connect two determinants that differ by more than a single occupation. By contrast, because of the double-excitation operator present in the two-electron transition-density elements... 

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