Editing Coupled cluster (section)

== Coupled-cluster equations ==

The Schrödinger equation can be written, using the coupled-cluster wave function, as

: <math>H |\Psi_0\rangle = H e^T |\Phi_0\rangle = E e^T |\Phi_0\rangle,</math>

where there are a total of ''q'' coefficients (''t''-amplitudes) to solve for. To obtain the ''q'' equations, first, we multiply the above Schrödinger equation on the left by <math>e^{-T}</math> and then project onto the entire set of up to {{not a typo|''m''-tuply}} excited determinants, where ''m'' is the highest-order excitation included in <math>T</math> that can be constructed from the reference wave function <math>|\Phi_0\rangle</math>, denoted by <math>|\Phi^*\rangle</math>. Individually, <math>|\Phi_i^a\rangle</math> are singly excited determinants where the electron in orbital ''i'' has been excited to orbital ''a''; <math>|\Phi_{ij}^{ab}\rangle</math> are doubly excited determinants where the electron in orbital ''i'' has been excited to orbital ''a'' and the electron in orbital ''j'' has been excited to orbital ''b'', etc. In this way we generate a set of coupled energy-independent non-linear algebraic equations needed to determine the ''t''-amplitudes:

: <math>\langle\Phi_0| e^{-T} H e^T |\Phi_0\rangle = E \langle\Phi_0|\Phi_0\rangle = E,</math>

: <math>\langle\Phi^*| e^{-T} H e^T |\Phi_0\rangle = E \langle\Phi^*|\Phi_0\rangle = 0,</math>

the latter being the equations to be solved, and the former the equation for the evaluation of the energy. (Note that we have made use of <math>e^{-T} e^T = 1</math>, the identity operator, and also assume that orbitals are orthogonal, though this does not necessarily have to be true, e.g., [[Valence bond theory|valence bond]] orbitals can be used, and in such cases the last set of equations are not necessarily equal to zero.)

Considering the basic CCSD method:

: <math>\langle\Phi_0| e^{-(T_1 + T_2)} H e^{(T_1 + T_2)} |\Phi_0\rangle = E,</math>
: <math>\langle\Phi_i^a| e^{-(T_1 + T_2)} H e^{(T_1 + T_2)} |\Phi_0\rangle = 0,</math>
: <math>\langle\Phi_{ij}^{ab}| e^{-(T_1 + T_2)} H e^{(T_1 + T_2)} |\Phi_0\rangle = 0,</math>

in which the similarity-transformed Hamiltonian <math>\bar{H}</math> can be explicitly written down using Hadamard's formula in Lie algebra, also called Hadamard's lemma (see also [[Baker–Campbell–Hausdorff formula]] (BCH formula), though note that they are different, in that Hadamard's formula is a lemma of the BCH formula):

: <math>\bar{H} = e^{-T} H e^{T} = H + [H, T] + \frac{1}{2!} \big[[H, T], T\big] + \dots = (H e^T)_C.</math>

The subscript ''C'' designates the connected part of the corresponding operator expression.

The resulting similarity-transformed Hamiltonian is non-Hermitian, resulting in different [[Eigenvalues and eigenvectors#Left and right eigenvectors|left and right vectors]] (wave functions) for the same state of interest (this is what is often referred to in coupled-cluster theory as the biorthogonality of the solution, or wave function, though it also applies to other non-Hermitian theories as well). The resulting equations are a set of non-linear equations, which are solved in an iterative manner. Standard quantum-chemistry packages ([[GAMESS (US)]], [[NWChem]], [[ACES (computational chemistry)|ACES II]], etc.) solve the coupled-cluster equations using the [[Jacobi method]] and direct inversion of the iterative subspace ([[DIIS]]) extrapolation of the ''t''-amplitudes to accelerate convergence.