An Eigenvalue Approach to Coupled Cluster Theory

Up to this point, our discussion has focused on the expansion of the wavefunction using the exponential ansatz given in Eq. [2.31]. When the cluster operator, $\hat{T}$, is truncated, the resulting CC wavefunction may be viewed as an approximate eigenfunction of the exact electronic Hamiltonian. However, another equally valid perspective focuses instead on construction of the exact eigenvectors of an approximate Hamiltonian. In configuration interaction theory, for example, one conventionally represents the electronic Hamiltonian within a determinantal basis consisting of the reference ($\Phi_0$), single excitations ($\Phi_i^a$), double excitations ( $\Phi_{ij}^{ab}$), etc. In the CISD approximation the Hamiltonian is represented schematically as

$$\hat{H}_{\rm CISD} = \left( \begin{array}{ccc} E_{\rm SCF} &... ...{H}_{D0} & \hat{H}_{DS} & \hat{H}_{DD} \\ \end{array}\right),$$ (3.18)

where $\hat{H}_{SD}$, for example, represents the block of Hamiltonian matrix elements between singly and doubly excited determinants and $E_{\rm SCF} = \langle \Phi_0 \vert \hat{H} \vert \Phi_0 \rangle$. We assume here that Brillouin's theorem[82] holds for the reference determinant, and therefore the matrix elements involving $\Phi_0$ and singly excited determinants are zero. The CISD energy is the lowest eigenvalue of this Hermitian matrix, and the CISD wavefunction is the corresponding eigenvector, i.e.,

$$\hat{H}_{\rm CISD} \vert \Psi_{\rm CISD} \rangle = E_{\rm CISD} \vert \Psi_{\rm CISD} \rangle.$$ (3.19)

The coupled cluster ``Schrödinger equation'', which leads to the energy and amplitude expressions given in Eqs. [3.9] and [3.10], may be written as

$$e^{-\hat{T}} \hat{H} e^{\hat{T}}\vert \Phi_0 \rangle = E \vert \Phi_0 \rangle.$$ (3.20)

Like Eq. [3.19], this equation represents an eigenvalue problem[101] in which the similarity-transformed Hamiltonian, $\bar{H} \equiv e^{-\hat{T}} \hat{H} e^{\hat{T}}$, is used in place of the bare electronic Hamiltonian, $\hat{H}$. The ground-state eigenvector of $\bar{H}$ is simply $\vert \Phi_0 \rangle$ with eigenvalue E. However, $\bar{H}$ is not Hermitian, unlike the CI Hamiltonian, and its matrix representation is therefore non-symmetric. In the CCSD approximation, for example,

$$\bar{H}_{\rm CCSD} = \left( \begin{array}{ccc} E_{\rm CCSD} ... ...D} \\ 0 & \bar{H}_{DS} & \bar{H}_{DD} \\ \end{array}\right),$$ (3.21)

where the CCSD energy is given by $\langle \Phi_0 \vert \bar{H} \vert \Phi_0 \rangle$, by Eq. [3.9] and $\bar{H}_{DS} \neq \bar{H}_{SD}$. The blocks of matrix elements $\langle \Phi_i^a \vert \bar{H} \vert \Phi_0 \rangle$ and $\langle \Phi_{ij}^{ab} \vert \bar{H} \vert \Phi_0 \rangle$ are both zero because the $\hat{T}$ amplitudes which parameterize the similarity transformation of $\hat{H}$ into $\bar{H}$ satisfy the equations,

$$0 = \langle \Phi_i^a \vert \bar{H} \vert \Phi_0 \rangle$$ (3.22)

and

$$0 = \langle \Phi_{ij}^{ab} \vert \bar{H} \vert \Phi_0 \rangle,$$ (3.23)

which are simply specific cases of Eq. [3.10]. Furthermore, unlike the CI case, $\bar{H}_{0S}$ is nonzero in spite of Brillouin's theorem because $\bar{H}$ includes contributions from products of the bare Hamiltonian with the cluster operators, $\hat{T}$.

As a result of the asymmetry of $\bar{H}$, the right-hand eigenvalue problem given in Eq. [3.20] is different from the left-hand eigenvalue problem,

$$\langle {\cal L} \vert \bar{H} = \langle {\cal L} \vert E.$$ (3.24)

The computed energy, E, however, is the same for both equations. In Eq. [3.24] above, the left eigenvector, $\langle {\cal L} \vert$, may be written in terms of a cluster operator, $\hat{{\cal L}}$, acting on the reference from the right, viz.

$$\langle {\cal L} \vert \equiv \langle \Phi_0 \vert \hat{{\cal L}}.$$ (3.25)

The operator $\hat{{\cal L}}$ may be defined in analogy to the cluster operator, $\hat{T}$, as a sum of of cluster operators,

$$\hat{{\cal L}} = 1 + \hat{{\cal L}}_1 + \hat{{\cal L}}_2 + \ldots.$$ (3.26)

The leading term of 1, which does not appear in $\hat{T}$ (cf. Eq. [2.29]), is required in order that the left- and right-hand eigenvectors have unit overlap with one another. Unlike the cluster operators, $\hat{T}_n$, the operators $\hat{{\cal L}}_n$ act to the left on $\langle \Phi_0 \vert$. Therefore, it is convenient to define them as de-excitation operators (or, equivalently, as bra-state excitation operators),

$$\hat{{\cal L}}_n = \left(\frac{1}{n!}\right)^2\sum_{ij\ldots ... ... a_{i}^{\dagger} a_{j}^{\dagger} \ldots a_{b}^{\ } a_{a}^{\ },$$ (3.27)

The task of determining the left-hand ground-state eigenvector of $\bar{H}$ is thus reduced to determining the amplitudes $l^{ij\ldots}_{ab\ldots}$. The ground-state coupled cluster energy may then be written as

$$E = \langle \Phi_0 \vert \hat{{\cal L}} \bar{H} \vert \Phi_0 \rangle,$$ (3.28)

where the left and right wavefunctions are assumed to be normalized according to $\langle \Phi_0 \vert \hat{{\cal L}} \vert \Phi_0 \rangle = 1$. This expression, which is more general than Eq. [3.9], provides a particularly useful starting point for the derivation of coupled cluster analytic energy derivatives; the left-hand eigenvector, $\langle \Phi_0 \vert \hat{{\cal L}}$, is related to the $\hat{\Lambda}$operator which arises due to the response of the cluster amplitudes to the external perturbation parameter.[49]

The concept of the coupled cluster method as an eigenvalue problem may be easily generalized to include excited states (in this case, states that are not the lowest in energy within a given symmetry). We may write the more general right-hand problem as

$$\bar{H} \hat{{\cal R}}(m) \vert \Phi_0 \rangle = E_m \hat{{\cal R}}(m) \vert \Phi_0 \rangle,$$ (3.29)

where

$$\hat{{\cal R}}(m) = \hat{{\cal R}}_0(m) + \hat{{\cal R}}_1(m) + \hat{{\cal R}}_2(m) + \ldots$$ (3.30)

represents a cluster operator expansion for the m-th excited state with energy E_m. For the ground state, $\hat{{\cal R}}(0) = 1$, as described above. Similarly, the left-hand eigenvalue problem becomes

$$\langle \Phi_0 \vert \hat{{\cal L}}(m) \bar{H} = \langle \Phi_0 \vert \hat{{\cal L}}(m) E_m.$$ (3.31)

``Biorthonormality'' of the left-hand and right-hand eigenvectors may be enforced such that

$$\langle \Phi_0 \vert \hat{{\cal L}}(m) \hat{{\cal R}}(n) \vert \Phi_0 \rangle = \delta_{mn},$$ (3.32)

leads to the generalized coupled cluster energy expression

$$E_m = \langle \Phi_0 \vert \hat{{\cal L}}(m) \bar{H} \hat{{\cal R}}(m) \vert \Phi_0 \rangle.$$ (3.33)

Note that the biorthonormality of the left- and right-hand states does not imply orthonormality of the left- or right-hand states among themselves, e.g.,

$$\langle \Phi_0 \vert \hat{{\cal R}}^{\dagger}(m) \hat{{\cal R}}(n) \vert \Phi_0 \rangle \neq \delta_{mn}.$$ (3.34)

The eigenvalue perspective described above does not offer any computational convenience for the ground-state problem because one must still use Eq. [3.10] to determine the cluster amplitudes that define the similarity transformation of the electronic Hamiltonian, $\hat{H}$, into the CC Hamiltonian, $\bar{H}$. However, this perspective does provide a rather simple CI-like approach for determining excited state wavefunctions. Equation-of-motion coupled cluster theory  (EOM-CC)[5,60,61,62,63,65], the name of which is based on early formulations involving response operators, has seen a considerable rise in popularity in recent years. The EOM-CCSD method[65,73], for example, is defined as the diagonalization of the CCSD effective Hamiltonian, $\bar{H}_{\rm CCSD}$ (where the cluster amplitudes are taken from the corresponding CCSD ground-state energy calculation) in the space of all singly and doubly excited determinants. It should be noted, however, that truncation of the cluster operator, $\hat{T}$, in the definition of $\bar{H}$ does not introduce errors into the EOM-CC energy, because the exact energy would still be obtained if the diagonalization basis were complete.

Much effort has been devoted recently to the development of a variety of excited-state coupled cluster techniques which are related to EOM-CC. For example, the linear-response coupled cluster (LR-CC) approach[73] originally described by Monkhorst[5] and recently implemented by several groups[69,102,103,70,,105] can be used to obtain identical results to those given by conventional EOM-CC. In addition, the symmetry-adapted cluster (SAC-CI) method devised independently by Nakatsuji[106,107,108] some years ago may be viewed as an approximation to EOM-CC and LR-CC. A relationship between EOM-CC and Fock-space multi-reference coupled cluster theory (FS-MRCC)[109,110,64,,112] has been exploited in the construction of methods for describing classes of doublet electronic states which are accessible via either electron-attachment (EOMEA-CC)[88,113] or ionization (EOMIP-CC)[109,110,111,67] from a given reference. Finally, we note the recent work by Nooijen and Bartlett on the similarity-transformed equation-of-motion coupled cluster (STEOM-CC) method[74,114], in which the effective Hamiltonian described above is further transformed using a reduced cluster operator, $\hat{S}$, which serves to decouple singly excited determinants from doubly and triply excited determinants in $\bar{H}$.