EOMXCC

EOMXCC A New Coupled-Cluster Method for Electronic Excited States... 

The purpose of this section is to introduce the basic equations of the new EOMXCC theory of electronically excited, ionized, and electron-attached states of molecular systems. As mentioned in the Introduction, the EOMXCC method is strongly linked to the so-called XCC theory, which applies to the ground-state problem. The specific variants of the EOMXCC formalism that apply to electron-attached states, ionized states, and electronically excited states are referred to as the EA-EOMXCC, IP-EOMXCC, and EE-EOMXCC methods, respectively. 

As in the ordinary EOMCC theory, in the EOMXCC method we solve the electronic Schrodinger equation (1) assuming that the excited states are represented by Eq. (7). We use the exponential representation of the ground-state wave function I S o), Eq. (8), but no longer assume that the cluster components Tn result from standard SRCC calculations (see below). The many-body expansions of the excitation operator Rk have the same form as in the ordinary EOMCC formalism. In particular, the three different forms of Rk discussed in the previous section [fi -E, R A, and REqs. (28), (30), and (26), respectively] are used to define the EE-EOMXCC, EA-EOMXCC, and IP-EOMXCC methods. As in the standard EOMCC method, by making suitable choices for the operators Qa, which define Rk, we can always extend the EOMXCC theory to other sectors of the Fock space. 

Equations (70) or (76) and (73) are the basic equations of the new EOMXCC theory. In order to solve an eigenvalue problem (70) we must decide about the source of information about the cluster operator T that defines H. We find the cluster amplitudes defining T by projecting Eq. (73) against the excited configurations included in the many-body expansion of... 

An eigenvalue problem characterizing the EOMXCC theory, Eq. (86), is similar to an eigenvalue problem characterizing the EOMCC formalism [cf. Eq. (33)]. The only essential difference is the type of the similarity transformed Hamiltonian used in both theories (H in EOMCC and H in EOMXCC). Similarity between the EOMCC and EOMXCC theories becomes even more transparent when we realize that we can replace the operator product Hu,open Ck,open in Eq. (86), by the explicitly connected expression Hu,open Ck, open)c>... 

Indeed, when Mr < Mt, the disconnected component of the left-hand side of Eq. (97), i.e. the expression P Ck,open Hn,oPen )i vanishes, since cluster amplitudes defining T, Eq. (41), satisfy equations (78) with n = 1,..., Mr-Equation (99) represents a generalization of the exact Eq. (88) to truncated EOMXCC schemes. Again, the only significant difference between the EOMXCC equations (98) and (99) and their EOMCC analogs (48) and (47), respectively, is the similarity transformed Hamiltonian used by both theories. As in the EOMCC theory, Eqs. (98) and (99) have the same general form (in particular, they rely on the same similarity transformed Hamiltonian) for all the sectors of Fock space. 

Since the cluster operator T used in the EOMXCC scheme is obtained by solving the XCC system of equations (78), matrix H representing the similarity transformed Hamiltonian Hn,open of the EOMXCC formalism, has a structure which is very similar to a structure of matrix representing... 

Hn0pen [cf. Eq. (50)]- Indeed, in the EE-EOMXCC theory, the matrix representing Hn,open has the form... 

Zeros in the first column of H are a consequence of the fact that the cluster operator T satisfies the XCC system of equations, Eq. (78). Their presence allows us to solve the EOMXCC eigenvalue problem in the space spanned by excited configurations only and obtain energy differences ujk directly. 

We do not have to determine the left-hand eigenstates of Hu if we are only interested in excitation energies. However, properties other than the energy need both left- and right-hand eigenstates of Hu- For example, the EOMXCC definition of transition moments involving the operator 0, which is compatible with expression (58), looks as follows,... 

Examples of truncated EOMXCC schemes are the IP-EOMXCCSD, EE-EOMXCCSD, and EA-EOMXCCSD methods, in which Mr = 2 and Mr < 2. In the most complete (Mr = 2) variant of the IP-EOMXCCSD method, where R1 is given by Eq. (62), the formula for the corresponding operator ClR becomes... 

A comparison of the most basic elements of the EOMCC and EOMXCC theories is given in Table 1. This table also shows basic equations of the Cl approach indicating the fact that the EOMCC and EOMXCC methods can be regarded as methods which are built on top of Cl (by replacing the Hamiltonian by its similarity transformed analogs). 

Table 1. A comparison of basic elements of Cl, EOMCC, and EOMXCC theories. 

As pointed out earlier, k in Eq. (154) cannot exceed 6 if T is approximated by its mono- and doubly excited components. Furthermore, we do not have to consider higher-than-four-body components of Hn,open at the CCSD level of the EOMXCC formalism. In fact, the only three- and four-body terms that we need in the EE-EOMXCCSD scheme are H and Hq, where P is defined by Eq. (134). These observations simplify our analysis, since very few three- and four-body contributions enter the projected Hamiltonian components H and Hf. In addition, many terms that enter Hz and Hq appear in higher orders of MBPT and can be neglected. 

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