.. include:: autodoc_abbr_options_c.rst

.. index::
     single: multireference
     single: Mk-MRCC

.. _`sec:psimrcc`:

PSIMRCC Implementation of Mk-MRCC Theory
========================================

.. codeauthor:: Francesco A. Evangelista and Andrew C. Simmonett 
.. sectionauthor:: Alexander E. Vaughn

*Module:* :ref:`Keywords <apdx:psimrcc>`, :ref:`PSI Variables <apdx:psimrcc_psivar>`, :source:`PSIMRCC <src/bin/psimrcc>`

State-specific Multireference coupled cluster theories provide highly
accurate energies and properties of electronic states that require
a multiconfigurational zeroth-order wavefunction.  The PSIMRCC
module contained in |PSIfour| implements the state-specific
multireference coupled-cluster approach of Mukherjee and co-workers
(Mk-MRCC). This method is implemented and shown to be a powerful tool in
[Evangelista:2006:154113]_ and [Evangelista:2008:124104]_. Mk-MRCC is
based on the Jeziorski-Monkhorst ansatz [Jeziorski:1981:1668]_ for the
wavefunction, :math:`\Psi`

.. math:: \left| \Psi \right \rangle = \sum_\mu^d e^{\hat{T}^\mu} \left| \Phi_\mu \right\rangle c_\mu \, \text{,}

where :math:`\Phi_\mu` are the reference determinants,
:math:`\hat{T}^\mu` are reference-specific excitation operators, and
:math:`c_\mu` are expansion coefficients obtained through diagonalization
of the Mk-MRCC effective Hamiltonian matrix that allows the various
reference determinants to interact. As an example of how this works
the Mk-MRCCSD excitation operators for each reference is contracted
two-body terms

.. math:: \hat{T}^\mu = \hat{T}^\mu_1 + \hat{T}^\mu_2

where

.. math:: \hat{T}^\mu_1 = \sum_i^{\textrm{occ}(\mu)} \sum_a^{\textrm{vir}(\mu)} t_i^a (\mu) \hat{a}^\dagger_a \hat{a}_i

and

.. math:: \hat{T}^\mu_2 =\frac{1}{4} \sum_i^{\textrm{occ}(\mu)} \sum_a^{\textrm{vir}(\mu)} t_{ij}^{ab} (\mu) \hat{a}^\dagger_b \hat{a}_j \hat{a}^\dagger_a \hat{a}_i  

The Mk-MRCC energy is a chosen eigenvalue of the effective Hamiltonian,
:math:`\textrm{H}^{eff}_{\mu \nu}`

.. math:: \sum_\nu \textrm{H}^{eff}_{\mu \nu} c_\nu =E c_\nu

where 

.. math:: \textrm{H}^{eff}_{\mu \nu} = \left \langle \Phi_\mu \right | \hat{H}e^{\hat{T}^\nu} \left | \Phi_\nu \right \rangle \, \textrm{.}

|PSIfour| currently implements Mk-MRCC with singles and doubles
[Mk-MRCCSD] and Mk-MRCCSD with perturbative triples [Mk-MRCCSD(T)]
as formulated in [Evangelista:2010:074107]_. A companion perturbation
method (Mk-MRPT2) has been developed based on the Mukherjee formalisim
as shown in [Evangelista:2009:4728]_.

The current version of the code is limited to reference active spaces
in which all determinants are connected to each other by no more than two
excitations.  In practice, this usually means that the active space can have
at most two particles, or at most two holes.  Examples would include 
CAS(2,2), CAS(2,8), CAS(4,3), etc., where CAS(n,m) refers to a
complete-active-space configuration interaction (CAS-CI) reference with n
electrons in m orbitals.  If the user specifies active spaces that do not fit
these limitations, then the code will still run, but some relevant
determinants will be missing, and the answer obtained will be an approximation
to the true Mk-MRCC procedure.

The PSIMRCC code itself does not perform orbital optimization.  Hence, the
references used might be considered CAS-CI references, but not CASSCF
references (CASSCF implies that the orbitals have been optimized specifically
to minimize the energy of the CAS-CI reference).  However, if one wishes to
use two-configuration self-consistent-field (TCSCF) orbitals, those can
be obtained using the multi-configuration self-consistent-field (MCSCF)
component of PSIMRCC (specifying |mcscf__reference| to be ``twocon``).
This is suitable for describing diradicals.  Otherwise, one may use RHF or
ROHF orbitals as input to PSIMRCC.  Due to a current limitation in the code,
one must obtain orbitals using PSIMRCC's MCSCF module regardless of what
orbital type is chosen, ``twocon``, ``rhf``, or ``rohf``.  An example of the
MCSCF input is given below.

PSIMRCC is most commonly used for low-spin cases (singlets or open-shell
singlets).  It is capable of performing computations on higher spin states
(e.g., triplets), but in general, not all the required matrix elements have
been coded for high-spin cases, meaning that results will correspond to an
approximate Mk-MRCC computation for high-spin cases.


A Simple Example
________________ 

The |psimrcc__corr_wfn| allows you to select one of three methods
Mk-MRPT2 [``PT2``], Mk-MRCCSD [``CCSD``], or Mk-MRCCSD(T) [``CCSD_T``].
The |psimrcc__corr_multp| option allows you to select the Slater
determinants with a particular :math:`M_s` value. The |psimrcc__wfn_sym|
keyword is neccesary if you do not want to compute the energy of the
totally-symmetric state. The |psimrcc__follow_root| option may be used
to follow different roots of the effective Hamiltonian. A value of 1
instructs PSIMRCC to follow the solution with the lowest energy given
a certain set of determinants. ::

   molecule o2 {
      0 3
      O
      O 1 2.265122720724
      units au
   }
   set {
      basis cc-pvtz
   }
   set mcscf {
      reference       rohf
      docc            [3,0,0,0,0,2,1,1]      # Doubly occupied MOs
      socc            [0,0,1,1,0,0,0,0]      # Singly occupied MOs
   }
   set psimrcc {
      corr_wfn        ccsd                   # Do Mk-MRCCSD 
      frozen_docc     [1,0,0,0,0,1,0,0]      # Frozen MOs
      restricted_docc [2,0,0,0,0,1,1,1]      # Doubly occupied MOs
      active          [0,0,1,1,0,0,0,0]      # Active MOs
      frozen_uocc     [0,0,0,0,0,0,0,0]      # Frozen virtual MOs
      corr_multp      1                      # Select the Ms = 0 component
      follow_root     1
      wfn_sym         B1g                    # Select the B1g state
   }
   energy('psimrcc')

Note that the oxygen molecule has 16 electrons (including core), while
the ``docc`` array contains only 7 doubly-occupied orbitals (or 14
electrons).  Hence, two more electrons are available to place into
the active space (given by ``active``), which consists of 2 orbitals.
Thus there are two active electrons in two orbitals.  In this particular
example, we are using standard ROHF orbitals for the Mk-MRCCSD procedure,
rather than TCSCF orbitals.  Nevertheless, with the present code, 
these orbitals must be provided through the MCSCF module, as specified in the
``set mcscf`` section above.

Orbital ordering and selection of the model space
_________________________________________________

The reference determinants :math:`\Phi_\mu` are specified in PSIMRCC
via occupational numbers. PSIMRCC requires that four arrays be specified
for this purpose.

- Frozen doubly occupied orbitals (|psimrcc__frozen_docc|) are doubly
  occupied in each reference determinant and are not correlated in the
  MRCC procedure.
- Doubly occupied orbitals (|psimrcc__restricted_docc|) are doubly
  occupied in each reference determinant and are correlated in the MRCC
  procedure.
- Active orbitals (|psimrcc__active|) are partially occupied in each
  reference determinant.
- Frozen virtual orbitals (|psimrcc__frozen_uocc|) are unoccupied in
  all reference determinants and are excluded from the correlated wave
  function.

The model space is selected by considering all possible occupations
of the electrons among the orbitals in the active space that result
in determinants with the correct symmetry (|psimrcc__wfn_sym|)
and the correct :math:`\textrm{M}_s` value specified by the keyword
|psimrcc__corr_multp|. Note that this does not consider the multiplicity
of the wavefunction. Thus, in order to obtain the wavefunction
with a set of :math:`\textrm{M}_s = 0` reference determinants for
an open-shell system you should request a |psimrcc__corr_multp| of
1 within the PSIMRCC module, and select the root of the effective
Hamiltonian that corresponds to the state of interest. In addition,
the |psimrcc__wfn_sym| keyword needs to be specified otherwise the
wavefunction belonging to the all-symmetric irrep will be selected. In
addition, it should be noted that for an open-shell singlet based
on two :math:`\textrm{M}_s = 0` determinants the eigenvector is
[:math:`\frac{1}{\sqrt{2}}\text{,}\frac{1}{\sqrt{2}}`], which corresponds
to a wavefunction of the following form:

.. math:: \frac{1}{\sqrt{2}} \left( \chi_1 \alpha (1) \chi_2 \beta (2) + \chi_2 \alpha(1) \chi_1 \beta (2) \right)

See Appendix :ref:`apdx:psimrcc` for a complete list of PSIMRCC options.
