EOM-CCSD

The EOM-CCSD method solves for excited states given a CCSD ground state. The key here is to apply a linear (CI-like) excitation operator to the CCSD ground state:¹

\[\psi=R e^{T_1+T_2}|\psi_0>\]

where \(R\) is given as:

\[ R = r_0 + \sum_{ia} r_{ia} \hat a^{\dagger}_a \hat a_i + \frac{1}{2} \sum_{ijab} r_{ijab} \hat a^{\dagger}_a \hat a_i \hat a^{\dagger}_b \hat a_j \]

EOM-CCSD is often quite accurate for excitation energies of singly-excited states when the underlying CCSD method for the ground state is reasonable (i.e., when CCSD breaks down for the groundstate because of bond-breaking, the EOM-CCSD method for excited states will also become suspect). Excited states which have significant double excitation character are often treated less accurately compared to those which are primarily of single excitation character.

To run an EOM-CCSD calculation in TeraChem one should start with a CIS calculation to obtain an initial guess for the \(R\) amplitudes, and of course a CCSD calculation for the ground state. A minimal example would include:

cis yes
cisnumstates 8
ccbox yes
ccbox_ccsd yes
ccbox_eomccsd_states 4
ccbox_eomccsd_guessvecs 8

That asks for a CIS calculation with 8 excited states, a CCSD calculation for the ground state, and an EOM-CCSD calculation of 4 excited states, using the 8 CIS states as an initial guess for the EOM-CCSD amplitudes. Setting ccbox_eomccsd_states to a positive integer is what triggers EOM-CCSD; setting it to 0 (the default) skips the EOM step.

EOM-CCSD currently produces singlet states only

ccbox_eomccsd_states requests singlet excited states. Triplet EOM-CCSD states are not currently available.

Worked example

A complete input that computes 4 EOM-EE-CCSD singlet states with a CIS-based guess and tight thresholds:

eom_ccsd_example.in

basis           cc-pvdz
sphericalbasis  false
coordinates     geom.xyz
method          rhf
run             energy

threall         1.0e-14
convthre        1.0e-10
xtol            1.0e-6
precision       double

charge          0
spinmult        1

cis                       yes
cisnumstates              8
cismax                    50
cismaxiter                100
cisguessvecs              20

ccbox                     yes
ccbox_ccsd                yes
ccbox_frozen_core         4
ccbox_scratch_dir         ./scr.geom
ccbox_cd_thresh           1.0e-8
ccbox_no_disk             yes
ccbox_ccsd_r_convthre     1.0e-8

ccbox_eomccsd_states      4
ccbox_eomccsd_maxiter     100
ccbox_eomccsd_r_convthre  1.0e-8
ccbox_eomccsd_guessvecs   8
ccbox_eomccsd_maxsubspace 28
end

The Davidson-style solver maintains a subspace of up to ccbox_eomccsd_maxsubspace guess vectors. As a rule of thumb, set the subspace size to roughly \(4 \times\) the number of requested states.

Custom initial-guess composition

ccbox_eomccsd_guessvecs is convenient for re-using CIS singles as the initial guess, but the underlying solver can also work from a custom mix of singles and doubles. Use the explicit keywords:

Keyword	Description
`ccbox_eomccsd_guess_singles`	Number of singly-excited CSFs in the initial guess (overrides `ccbox_eomccsd_guessvecs`)
`ccbox_eomccsd_guess_doubles`	Number of doubly-excited CSFs in the initial guess

This is the right knob to reach for if Davidson is having trouble locating a state with significant double-excitation character: increasing ccbox_eomccsd_guess_doubles enriches the initial subspace with double excitations.

Transition properties

Setting ccbox_properties yes activates one-electron property calculation, which gives the ground-state dipole moment and transition dipole moments between EOM-CCSD states. Setting ccbox_ntos yes (which also requires ccbox_properties yes) additionally writes the natural transition orbitals for each transition.

EOM-IP-CCSD

Ionization potentials can be computed with the EOM-IP variant, in which the excitation operator \(R\) removes an electron rather than promoting one. It is turned on by ccbox_eomccsd_ip_states <N> and is otherwise configured by a parallel family of ccbox_eomccsd_ip_* keywords:

eom_ip_ccsd_example.in

basis           cc-pvdz
sphericalbasis  true
coordinates     geom.xyz
method          rhf
run             energy

threall         1.0e-14
convthre        1.0e-10
xtol            1.0e-6
precision       double

charge          0
spinmult        1

ccbox                          yes
ccbox_ccsd                     yes
ccbox_scratch_dir              ./scr.geom
ccbox_cd_thresh                1.0e-8
ccbox_no_disk                  yes
ccbox_ccsd_r_convthre          1.0e-8

ccbox_eomccsd_ip_states        5
ccbox_eomccsd_ip_maxiter       100
ccbox_eomccsd_ip_r_convthre    1.0e-8
ccbox_eomccsd_ip_guessvecs     10
ccbox_eomccsd_ip_maxsubspace   50
end

Core-valence-separated variants of both EOM-EE and EOM-IP (useful for core-excited spectra) are available through ccbox_eomccsd_cvs_states and ccbox_eomccsd_ip_cvs_states respectively, with analogous companion keywords.

Summary of relevant keywords

EOM-EE-CCSD (excitation energies, singlets)

Keyword	Type	Default	Description
ccbox_eomccsd_states	integer	0	Number of EOM-EE-CCSD excited states. A positive value triggers the EOM calculation.
ccbox_eomccsd_maxiter	integer	100	Maximum Davidson iterations
ccbox_eomccsd_r_convthre	float	1.0e-6	Amplitude convergence threshold
ccbox_eomccsd_guessvecs	integer	1	Number of singly-excited guess vectors (alias for `ccbox_eomccsd_guess_singles`)
ccbox_eomccsd_guess_singles	integer	= `ccbox_eomccsd_states`	Number of singly-excited guess vectors
ccbox_eomccsd_guess_doubles	integer	0	Number of doubly-excited guess vectors
ccbox_eomccsd_maxsubspace	integer	4 × `ccbox_eomccsd_states`	Maximum Davidson subspace size
ccbox_eomccsd_algorithm	string	`block`	Davidson algorithm: `block`, `single`, or `mixed`
ccbox_properties	bool	no	Compute ground-state dipole and EOM transition dipoles
ccbox_ntos	bool	no	Compute natural transition orbitals (requires `ccbox_properties yes`)

EOM-IP-CCSD (ionization potentials)

Keyword	Type	Default	Description
ccbox_eomccsd_ip_states	integer	0	Number of EOM-IP-CCSD states
ccbox_eomccsd_ip_maxiter	integer	—	Max Davidson iterations
ccbox_eomccsd_ip_r_convthre	float	—	Amplitude convergence threshold
ccbox_eomccsd_ip_guessvecs	integer	—	Number of guess vectors
ccbox_eomccsd_ip_maxsubspace	integer	4 × `ccbox_eomccsd_ip_states`	Max Davidson subspace size

Core-valence separation variants

Keyword	Description
ccbox_eomccsd_cvs_states	Number of core-valence-separated EOM-EE-CCSD states
ccbox_eomccsd_ip_cvs_states	Number of core-valence-separated EOM-IP-CCSD states

For Cholesky decomposition and general ccbox_* keywords, see the Coupled Cluster overview.

References

J. F. Stanton and R. J. Bartlett, J. Chem. Phys. 98 7029 (1993) ↩