Coupled Cluster

In this section the working equations for Coupled Cluster can be found.

Coupled Cluster Singles Double

For CCSD the T1 (tia) and T2 (tijab) amplitudes is constructed to calculate the energy. These are constructed by the fisrt creating some intermediates:

\[F_{ae}=\left(1-\delta_{ae}\right)f_{ae}-\frac{1}{2}\sum_{m}^{occ}f_{me}t_{m}^{a}+\sum_{m,f}^{occ,virt}t_{m}^{f}\left\langle ma\left|\right|fe\right\rangle -\frac{1}{2}\sum_{m,n,f}^{occ,occ,virt}\tilde{\tau}_{mn}^{af}\left\langle mn\left|\right|ef\right\rangle\]
\[F_{mi}=\left(1-\delta_{mi}\right)f_{mi}+\frac{1}{2}\sum_{e}^{virt}t_{i}^{e}f_{me}+\sum_{e,n}^{virt,occ}\left\langle mn\left|\right|ie\right\rangle +\frac{1}{2}\sum_{n,e,f}^{occ,virt,virt}\tilde{\tau}_{in}^{ef}\left\langle mn\left|\right|ef\right\rangle\]
\[F_{me}=f_{me}+\sum_{n,f}^{occ,virt}t_{n}^{f}\left\langle mn\left|\right|ef\right\rangle\]
\[W_{mnij}=\left\langle mn\left|\right|ij\right\rangle +P_{-}\left(ij\right)\sum_{e}^{virt}t_{j}^{e}\left\langle mn\left|\right|ie\right\rangle +\frac{1}{4}\sum_{e,f}^{virt,virt}\tau_{ij}^{ef}\left\langle mn\left|\right|ef\right\rangle\]
\[W_{abef}=\left\langle ab\left|\right|ef\right\rangle -P_{-}\left(ab\right)\sum_{m}^{occ}t_{m}^{b}\left\langle ma\left|\right|ef\right\rangle +\frac{1}{4}\sum_{m,n}^{occ,occ}\tau_{mn}^{ab}\left\langle mn\left|\right|ef\right\rangle\]
\[W_{mbej}=\left\langle mb\left|\right|ej\right\rangle +\sum_{f}^{virt}\left\langle mb\left|\right|ef\right\rangle -\sum_{n}^{occ}t_{n}^{b}\left\langle mn\left|\right|ej\right\rangle -\sum_{n,f}^{occ,virt}\left(\frac{1}{2}t_{jn}^{fb}+t_{j}^{f}t_{n}^{b}\right)\left\langle mn\left|\right|ef\right\rangle\]

In the above equations the following definitions is used:

\[\tilde{\tau}_{ij}^{ab}=t_{ij}^{ab}+\frac{1}{2}\left(t_{i}^{a}t_{j}^{b}-t_{i}^{b}t_{j}^{a}\right)\]
\[\tau=t_{ij}^{ab}+t_{i}^{a}t_{j}^{b}-t_{i}^{b}t_{j}^{a}\]
\[P_{-}\left(ij\right)=1-P\left(ij\right)\]

The T1 and T2 is the constructed as:

\[ \begin{align}\begin{aligned}t_{i}^{a}D_{i}^{a}=f_{ia}+\sum_{e}^{occ}t_{i}^{e}F_{ae}-\sum_{m}^{occ}t_{m}^{a}F_{mi}+\sum_{m,e}^{occ,virt}t_{im}^{ae}F_{me}-\sum_{n,f}^{occ,virt}t_{n}^{f}\left\langle na\left|\right|if\right\rangle\\-\frac{1}{2}\sum_{m,e,f}^{occ,virt,virt}t_{im}^{ef}\left\langle ma\left|\right|ef\right\rangle -\frac{1}{2}\sum_{m,e,n}^{occ,virt,occ}t_{mn}^{ae}\left\langle mn\left|\right|ei\right\rangle\end{aligned}\end{align} \]
\[ \begin{align}\begin{aligned}t_{ij}^{ab}D_{ij}^{ab}=\left\langle ij\left|\right|ab\right\rangle +P_{-}\left(ab\right)\sum_{e}^{virt}t_{ij}^{ae}\left(F_{be}-\frac{1}{2}\sum_{m}^{occ}t_{m}^{b}F_{me}\right)+\frac{1}{2}\sum_{e,f}^{virt,virt}\tau_{ij}^{ef}W_{abef}\\-P_{-}\left(ij\right)\sum_{m}^{occ}t_{im}^{ab}\left(F_{mj}+\frac{1}{2}\sum_{e}^{virt}t_{j}^{e}F_{me}\right)+\frac{1}{2}\sum_{m,n}^{occ,occ}\tau_{mn}^{ab}W_{mnij}\\+P_{-}\left(ij\right)P_{-}\left(ab\right)\sum_{m,e}^{occ,virt}\left(t_{im}^{ae}W_{mbej}-t_{i}^{e}t_{m}^{a}\left\langle mb\left|\right|ej\right\rangle \right)\\+P_{-}\left(ij\right)\sum_{e}^{virt}t_{i}^{e}\left\langle ab\left|\right|ej\right\rangle -P_{-}\left(ab\right)\sum_{m}^{occ}t_{m}^{a}\left\langle mb\left|\right|ij\right\rangle\end{aligned}\end{align} \]

Here:

\[D_{i}^{a}=f_{ii}-f_{aa}\]
\[D_{ij}^{ab}=f_{ii}+f_{jj}-f_{aa}-f_{bb}\]

It can be noted that the T1 and T2 equations depends on T1 and T2. Thus it have to be solver iteratively. The initial guess is given as:

\[t_{i}^{a}=0\]
\[t_{ij}^{ab}=\frac{\left\langle ij\left|\right|ab\right\rangle }{D_{ij}^{ab}}\]

The CCSD energy is then found as:

\[E_{\mathrm{CCSD}}=\sum_{i,a}^{occ,virt}f_{ia}t_{i}^{a}+\sum_{i,j,a,b}^{occ,occ,virt,virt}\left\langle ij\left|\right|ab\right\rangle \left(\frac{1}{4}t_{ij}^{ab}+\frac{1}{2}t_{i}^{a}t_{j}^{b}\right)\]

FUNCTION:

  • CC.CCSD(occ, F, C, VeeMOspin, maxiter, deTHR, rmsTHR, runCCSDT=0)
  • return EMP2, ECCSD

Input:

  • F, fock matrix in spatial basis
  • C, MO coeffcients in spatial basis
  • VeeMOspin, two electron integrals in spinbasis
  • deTHR, change in energy check for convergence
  • rmsTHR, check for change in T1 and T2
  • runCCSDT, 0 for CCSD and 1 for CCSD(T)

Output:

  • EMP2, MP2 energy
  • ECCSD, CCSD energy

References:

Perturbative Triples Correction

To find the perturbative triples correction, the disconnected and connected T3 have to be calculated. The disconnected is found as:

\[D_{ijk}^{abc}t_{ijk,\mathrm{disconnected}}^{abc}=P\left(i/jk\right)P\left(a/bc\right)t_{i}^{a}\left\langle jk\left|\right|bc\right\rangle\]

And the connected is found as:

\[D_{ijk}^{abc}t_{ijk,\mathrm{connected}}^{abc}=P\left(i/jk\right)P\left(a/bc\right)\left[\sum_{e}^{virt}t_{jk}^{ae}\left\langle ei\left|\right|bc\right\rangle -\sum_{m}^{occ}t_{im}^{bc}\left\langle ma\left|\right|jk\right\rangle \right]\]

In the above equations the following definitions is used:

\[D_{ijk}^{abc}=f_{ii}+f_{jj}+f_{kk}-f_{aa}-f_{bb}-f_{cc}\]
\[P\left(i/jk\right)f\left(i,j,k\right)=f\left(i,j,k\right)-f\left(j,i,k\right)-f\left(k,j,i\right)\]

The energy correction can now be found as:

\[E_{\mathrm{\left(T\right)}}=\frac{1}{36}\sum_{i,j,k,a,b,c}^{occ,occ,occ,virt,virt,virt}t_{ijk,\mathrm{connected}}^{abc}D_{ijk}^{abc}\left(t_{ijk,\mathrm{connected}}^{abc}+t_{ijk,\mathrm{disconnected}}^{abc}\right)\]

FUNCTION:

  • see CCSD function above
  • return EMP2, ECCSD, ET

Input:

  • runCCSDT=1

Output:

  • EMP2, MP2 energy
  • ECCSD, CCSD energy
  • ET, perturbative triples corrections

References: