Features

The eT program is first and foremost a coupled cluster program, with the CCS, CC2, CCSD, CCSD(T), and CC3 methods implemented. In addition to the standard coupled cluster methods, eT has multilevel CC2, multilevel CCSD, and coupled cluster time propagation. Besides coupled cluster methods, MP2, FCI and TDHF are implemented. Hartree-Fock calculations, both restricted closed shell, restricted open-shell, and unrestricted, are available. Furthermore, the multilevel Hartree-Fock method is implemented. QM/MM is available at both HF and CC level.

A detailed description of the features available for each method is given below.

Hartree-Fock

The available Hartree-Fock methods are

  • Restricted Hartree-Fock (RHF, ROHF)

  • Unrestricted Hartree-Fock (UHF, CUHF)

  • Multilevel Hartree-Fock (MLHF)

  • Quantum electrodynamics Hartree-Fock (QED-HF)

  • Strong coupling quantum electrodynamics Hartree-Fock (SC-QED-HF)

  • Localization of Hartree-Fock orbitals (Edmiston-Ruedenberg, Foster-Boys)

Restricted and unrestricted Hartree-Fock

Restricted closed-shell Hartree-Fock (RHF) can be used for single-point calculations, geometry optimizations, properties (dipole and quadrupole), population analysis (Mulliken and/or Loewdin), and as a reference wave function for coupled cluster calculations. RHF can be used within the Cavity Born-Oppenheimer approximation for single-point calculations.

Restricted open-shell Hartree-Fock (ROHF) can be used for single-point calculations.

Unrestricted Hartree-Fock (UHF) can be used for single-point calculations.

Setting up a Hartree-Fock calculation

Multilevel Hartree-Fock

Multilevel Hartree-Fock can be used for single-point calculations, and as a reference wave function for reduced space coupled cluster calculations.

Setting up a multilevel Hartree-Fock calculation

Quantum electrodynamics Hartree-Fock

Quantum electrodynamics Hartree-Fock supports the same features as restricted closed-shell Hartree-Fock (RHF) including molecular gradients, but the current geometry optimization does not work.

Strong coupling Quantum electrodynamics Hartree-Fock

Strong coupling quantum electrodynamics Hartree-Fock (SC-QED-HF) is available for calculations of ground state energies.

Time-dependent Hartree-Fock

Excitation energies and oscillator strengths at the TDHF level of theory are available either within the random phase (RPA) or the Tamm-Dancoff approximation (TDA). In addition, static and frequency-dependent polarizabilities can be obtained.

Quantum electrodynamics time-dependent Hartree-Fock

Quantum electrodynamics time-dependent Hartree-Fock supports the same features as time-dependent Hartree-Fock except the polarizabilities. In addition, a measure of the photon character of each excitation is computed.

Coupled cluster

The implemented coupled cluster methods in eT are

  • Standard methods: CCS and CCSD, CCSDT

  • Perturbative methods: CC2, CC3 and CCSD(T)

  • Multilevel CC2 and CCSD

  • Quantum electrodynamics coupled cluster: QED-CCSD

Additionally, the code can perform time-propagation of some of the implemented coupled cluster wave functions.

The features implemented for the coupled cluster methods vary somewhat, and are detailed below.

Setting up a coupled cluster calculation

CCS, CC2, CCSD, CC3 and CCSDT

For CCS, CC2, CCSD, and CC3, eT offers ground and excited state calculations in addition to dipole and quadrupole moments, EOM oscillator strengths and EOM Dyson orbitals.

Core-excited states for these methods are available employing core-valence separation (CVS). Ionized and core-ionized states can be computed using the diffuse orbital trick and CVS.

Core-excited states for these methods are available employing core-valence separation (CVS). Ionized and core-ionized states can be computed using the diffuse orbital trick and CVS.

For CCS, CC2, and CCSD also EOM polarizabilities and linear response oscillator strengths as well as polarizabilities are implemented. It is also possible to carry out the EOM polarizability calculations based on an asymmetric expression.

For CCS, CC2, and CCSD triplet excitation energies are implemented.

For CCSDT, ground and excited state energies are implemented.

Ground state energies for all CC methods in the Cavity Born-Oppenheimer approximation are implemented.

Properties

Currently, EOM oscillator strengths are available in length, velocity and mixed gauge, and EOM rotatory strengths are available in length and velocity gauge for CCS, CC2, CCSD and CC3.

CCSD(T)

Ground state energies at the CCSD(T) level of theory are implemented.

Low-memory CC2

In eT, there are two versions of the CC2 code. The standard CC2 code has a memory requirement proportional to \(n_o^2 n_v^2\) , where \(n_o\) is the number of occupied orbitals and \(n_v\) is the number of virtual orbitals. However, the low-memory CC2 implementation has an \(N^2\) memory requirement, where \(N\) is the number of orbitals.

Currently, only ground and excited state energies are available with the low-memory CC2 code.

Multilevel coupled cluster

The multilevel CC2 (MLCC2) and multilevel CCSD (MLCCSD) methods are available with correlated natural transition orbitals, Cholesky orbitals (occupied and virtual), and projected atomic orbitals.

Currently, only ground and excited states are available at the MLCCSD level of theory.

Setting up a multilevel coupled cluster calculation

Time-dependent coupled cluster

Real-time propagation is available for CCS and CCSD methods. It solves the differential equations describing the time evolution of cluster amplitudes and Lagrange multipliers. The available integrators are

  • Euler

  • Gauss-Legendre (with order 2, 4 and 6)

  • Runge-Kutta (with order 4)

  • Dormand-Prince (with orders 5(4) and 8(5,3))

The time dependent quantities that are available as output are:

  • Amplitudes

  • Multipliers

  • MO density matrix

  • Energy

  • Electric field

  • Dipole moment

In addition, visualizable time-dependent density and spectra given by the Fast Fourier Transform of the dipole moment and of the electric field can be requested as output.

Quantum electrodynamics coupled cluster

Ground and excited state energies at the QED-CCSD level are implemented.

Warning

Excitation energies are only obtainable by requesting left eigenvectors in the solver cc es section.

Cholesky decomposition of the ERIs

The coupled cluster code is based on Cholesky decomposed electron repulsion integrals (ERIs). For sufficiently small systems, T1-transformed ERIs are constructed from the Cholesky vectors and stored in memory. For systems where the ERIs cannot be placed in memory, the T1-transformed Cholesky vectors are stored in memory if possible and ERIs are constructed from these vectors on the fly. For larger systems, the Cholesky vectors are stored on disk.

Other methods

FCI

Full CI calculations are available for both closed (RHF reference) and open (ROHF reference) shells. In addition, dipole and quadrupole moments can be calculated for the ground state. Ground state energies can be calculated within the Cavity Born-Oppenheimer approximation.

QED-FCI

QED Full CI calculations are available for both closed (QED-HF or RHF reference) and open (ROHF or UHF reference) shells. In addition, dipole and quadrupole moments can be calculated for the ground state. Also transition dipole and quadrupole moments can be calculated.

MP2

Ground state energies at the MP2 level of theory are implemented.

Environments

The available QM/MM methods are

  • Electrostatic embedding (non-polarizable)

  • Polarizable QM/Fluctuating charges

  • Polarizable Continuum Model

These methods can be coupled both with HF or CC wave functions. In case of QM/FQ, only HF ground state Fock and MOs are affected by QM/FQ.

Ground state energy calculations within the Cavity Born-Oppenheimer approximation can be run for HF, CC and CI. The photonic degrees of freedom are treated as a non-polarizable embedding.

Geometry optimization and frequencies

Vibrational frequencies, normal modes, and Wigner sampling is available at the HF (ground state) and CC levels of theory (ground and excited states), obtained from numerical differentiation to approximate the nuclear Hessian. This is currently efficiently implemented for HF and CCSD, where analytical gradients are available.

Ground state

Analytical gradients are available at the HF and CCSD level of theory. Numerical gradients are available for HF and all CC methods.

Excited state

Analytical gradients are available at the CCSD level of theory. Numerical gradients are available for all CC methods.

Visualization

Ground state densities can be written to .plt or .cube files that are readable by Chimera for both Hartree-Fock and coupled cluster. For coupled cluster, it is also possible to plot transition densities, natural transition orbitals, and Dyson orbitals.