Specific Walkthroughs
Here, we walkthrough some code for performing a few of the standard
CC calculations that are possible using CCpy. In all cases, we will
assume that the Driver object has been instantiated, either
manually, or using one of the provided interfaces. This way, the
system is specified within driver.system and the molecular
orbital integrals are loaded and available in driver.hamiltonian.
Completely Renormalized (CR) CC Calculations
Here is an example of how to perform a CR-CC(2,3) calculation in order to obtain the nonperturbative correction to the CCSD energetics for the effects of missing triples using moment energy expansions.
# Run CCSD calculation
driver.run_cc(method="ccsd")
# Obtain CCSD similarity-transformed Hamiltonian
driver.run_hbar(method="ccsd")
# Run left-CCSD calculation
driver.run_leftcc(method="left_ccsd")
# Run CR-CC(2,3) triples correction
driver.run_ccp3(method="crcc23")
The above commands, which must be run in sequence, first performs the CCSD calculation in order
to obtain the converged \(T_{1}\) and \(T_{2}\) clusters and follows this
by computing the one- and two-body components of the CCSD similarity-transformed
Hamiltonan \(\bar{H}=(H_N e^{T_1+T_2})_C\). Note that the run_hbar
method replaces the bare one- and two-body molecular orbital integrals contained
in driver.hamiltonian with their similarity-transformed counterparts.
Next, the driver.run_leftcc(method="left_ccsd") method solves the companion left-CCSD system of
linear equations to obtain the \(\Lambda_{1}\) and \(\Lambda_{2}\) operators,
which are needed to compute the CR-CC(2,3) triples correction using a single
noniterative step with driver.run_ccp3(method="crcc23"). The CR-CC(2,3)
calculation returns four distinct energetics, labelled as CR-CC(2,3)X, for
X = A, B, C, and D, where each variant A-D corresponds to a different treatment of the energy
denominator \(E^{(\text{CCSD})} - \langle \Phi_{ijk}^{abc} | \bar{H} | \Phi_{ijk}^{abc} \rangle\)
entering the formula for the CR-CC(2,3) triples correction. Note that the variant using
the simplest Moeller-Plesset form of the energy denominator, denoted as CR-CC(2,3)A,
is equivalent to the method called CCSD(2)T and the
result corresponding to CR-CC(2,3)D, which employs the full Epstein-Nesbet
energy denominator, is generally most accurate and often simply referred to as the
CR-CC(2,3) energy (or by its former name, CR-CCSD(T)L).
Active-Orbital-Based CC(P; Q) Calculations
Here is an example of how to perform a CC(t;3) calculation in order to obtain the nonperturbative correction to the CCSDt energetics for the effects of missing triples using the CC(P;Q) moment expansions.
# Set number of occupied and unoccupied orbitals that are active
nacto = 2
nactu = 2
driver.system.set_active_space(nact_occupied=nacto, nact_unoccupied=nactu)
# Run active-space CCSDt calculation
driver.run_cc(method="ccsdt1")
# Obtain CCSD-like similarity-transformed Hamiltonian
driver.run_hbar(method="ccsd")
# Run left-CCSD calculation
driver.run_leftcc(method="left_ccsd")
# Run CC(t;3) correction
driver.run_ccp3(method="cct3")
Compared to the above code used to perform CR-CC(2,3), the main difference in CC(t;3) is that the lower-order CCSD
calculation is replaced with the improved active-space CCSDt approach. In order to run
CCSDt, a contiguous set of active orbitals around the Fermi level must be specified using
the driver.system.set_active_space method. Here, we are arbitrarily setting up a (2,2)
active space, but the reasonable choice of active orbitals is obviously very dependent on the
specifics of the problem in consideration. This particular example of CC(t;3) relies on what
is known as the two-body approximation, which employs the CR-CC(2,3)-like expressions to formulate
the correction for missing triples to the CCSDt energetics. As a result, the components of the
similarity-transformed Hamiltonian as well as the left-CC deexcitation operator are obtained
with CCSD-like calculations that avoid directly using the of active-orbital-based \(t_3\)
clusters in the resulting expressions, and instead indirectly incorporates their effects
by using \(T_1\) and \(T_2\) clusters provided by CCSDt, which are
relaxed in the presence of \(t_3\), and are more accurate than the one- and two-body
components of the cluster operator obtained in CCSD calculations.
Using the more general CC(P) and CC(P;Q) routines, one can also perform the most complete CC(t;3) correction that does not invoke the two-body approximation.
# Set number of occupied and unoccupied orbitals that are active
nacto = 2
nactu = 2
driver.system.set_active_space(nact_occupied=nacto, nact_unoccupied=nactu)
# Import and use the helper routine to build the P space corresponding to the choice of active orbitals
from ccpy.utilities.pspace import get_active_space
t3_excitations, pspace = get_active_pspace(driver.system, target_irrep=driver.system.reference_symmetry)
# Run the active-space CCSDt calculation using the general CC(P) solver
driver.run_ccp(method="ccsdt_p", t3_excitations=t3_excitations)
# Obtain the CCSDt similarity-transformed Hamiltonian
driver.run_hbar(method="ccsdt_p", t3_excitations=t3_excitations)
# Run the left-CCSDt calculation
driver.run_leftccp(method="left_ccsdt_p", t3_excitations=t3_excitations)
# Run the complete CC(t;3) correction
driver.run_ccp3(method="ccp3", state_index=0, t3_excitations=t3_excitations, two_body_approx=False)
Here, the list of triply excited determinants that enter the P space corresponding to the choice of a (2,2)
orbital active space is constructed with the help of the get_active_pspace function, returning
a list of spin-orbital triples excitations stored in the 2D Numpy array called t3_excitations. This
list of triply excited determinants entering the P space can also be constrained to include only those belonging
to a specific symmetry irrep, which is specified using the keyword target_irrep.
For ground-state calculations of molcules described using an Abelian point group, the target irrep
should correspond to the symmetry of the reference wave function, which is stored in the attribute
driver.system.reference_symmetry (for closed-shell systems, this should correspond to the totally
symmetric representation, however, this may not be the case for open shells). Using the P space containing
all singles and doubles ad the list of triples excitations contained in t3_excitations, the CC(P)
calculations can be performed using the general driver.run_ccp method. This is followed by the running
the driver.run_hbar and driver.run_leftccp routines, which are used to compute the CCSDt
similarity-transformed Hamiltonian and solve the left-CCSDt system of linear equations, respectively.
Finally, the complete CC(t;3) correction is computed using the driver.run_ccp3 routine, where it
is important to set the flag two_body_appox=False (by default, it is set to True). Unlike
the CC(t;3) calculations employing the two-body approximation, which takes advantage of CCSD-like routines
that incorporate the effects of the active-space \(t_3\) clusters indirectly through relaxation of the
\(T_1\) and \(T_2\) components, the full CC(t;3) correction requires constructing the corresponding
CCSDt similarity-transformed Hamiltonian using the \(T_1\), \(T_2\), and \(t_3\) operators and
solving for the companion one-, two-, and three-body components of the left-CCSDt deexcitation operator.
As a result, the full CC(t;3) correction provides energetics that are more accurate than its counterpart
employing the two-body approximation, but with computational costs that are larger by roughly a factor of
2-3. For excited-state applications of CC(t;3), the improvement in the energetics often justifies
this increase in computational effort. On the other hand, the two-body approximation is often sufficient for
ground-state CC(t;3) calaculations, with previous applications demonstrating its ability to converge the parent
CCSDT energetics to within a fraction of a millihartree, even in challenging situations featuring stronger
many-electron correlation effects where \(T_3\) clusters become large and nonperturbative.