The ContractedSchrodinger command computes the ground-state energy of a molecule from solving a contracted Schrödinger equation (CSE) for the two-electron reduced density matrix (2-RDM).   The command solves the anti-Hermitian part of the CSE (ACSE) for the direct determination of the 2-RDM without the many-electron wave function.

The initial guess for the 2-RDM for the solution of the ACSE is computed by a complete-active-space (CAS) calculation using the orbitals and electrons set by the active keyword.  The active keyword is a list of two elements [N,r] containing the number N of active electrons and the number r of active orbitals or a set {} containing the indices of the molecular orbitals to be treated as active.  If the active keyword is not assigned, then a [2,2] active space is automatically chosen for even N and a [3,3] active space for odd N.   

The CAS calculation employs one of the following three methods: ActiveSpaceSCF (default), ActiveSpaceCI, or Variational2RDM.  The CAS method can be changed with the optional keyword casmethod by assigning casmethod to one of the three procedure names: ActiveSpaceSCF, ActiveSpaceCI, or Variational2RDM.  (Note that the full name QuantumChemistry:-Variational2RDM is required if the Quantum Chemistry Package has not been loaded with the with command).

The ACSE method is applicable to both ground and excited excited states with the state being selected by assigning the optional keyword state to a nonnegative integer with 0 being the ground state (default) and 1 being the first excited state.  

The ACSE method is also applicable to any total spin state S with the total spin being set by assigning the optional keyword spin to the nonnegative integer 2S with 0 being S=0 and 1 being S=1/2.

Because the ACSE adds electron correlation to the 2-RDM from the CAS calculation, it is a multi-reference 2-RDM method in which the reference is not represented by a wave function but by a 2-RDM.  For a mutireference method It has a low computational scaling of r_a²r_e⁴ where r_a is the rank of the active orbitals and r_e is the rank of the external (core and virtual) orbitals.

For low-to-moderate electron correlation the ACSE has an accuracy approaching that of coupled cluster with single, double, and triple excitations; however, unlike the single-reference coupled cluster methods, the ACSE can accurately describe the energies and properties of strongly correlated ground and excited states.

The efficiency of the ACSE can be increased by using the optional keyword active_cse to specify a second, larger active space.  When this keyword is given, the ACSE solution is computed in this second active space with the remaining orbitals frozen.   

The use of the Variational2RDM method for the CAS solver is especially useful for large CAS spaces that cannot be treated by traditional CASCI and CASSCF calculations that rely upon a configuration interaction calculation that grows exponentially with the size of the active space.  For casmethod=Variational2RDM, the optional casscf keyword controls whether or not the orbitals of the active space are optimized.  If set to false (default), the orbitals are not optimized, and if set to true, the orbitals are optimized.    

Both the spin-free 1- and 2-RDMs are returned from the ACSE calculation.

active = list or set -- [number of electrons, number of active orbitals] or {integer indices of the active orbitals}

active_cse = list -- [number of electrons in ACSE, number of active orbitals in ACSE]

basis = string -- name of the basis set.  See Basis for a list of available basis sets.  Default is "sto-3g".

casmethod = proc -- name of the CAS method: ActiveSpaceSCF (default), ActiveSpaceCI, or Variational2RDM.  

casscf = boolean -- (only for casmethod=Variational2RDM) optimize the active orbitals through rotations with the inactive orbitals. Default is false.

charge = nonnegint -- net charge of the molecule. Default is 0.

conditions = string -- (only for casmethod=Variational2RDM) "D", "DQ", "DQG," or "DQGT". Default is "DQG".

conv_tol_cse = float -- converge threshold. Default is 1.0*${10}^{-7}\.$

ghost = list of lists -- each list has the string of an atom's symbol and the atom's x, y, and z coordinates.  See Ghost Atoms.

initial_mo = list -- initial molecular orbitals (MOs) as a list: [ t[mo_coeff], t[mo_symmetry] ] where t[mo_coeff] is the Matrix of MOs (columns) in terms of atomic orbitals (rows) and t[mo_symmetry] is the Vector of MO symmetries (see HartreeFock output).

max_cycle = posint -- max number of iterations. Default is 100.

max_memory = posint/boolean -- allowed memory in MB. Default is 4000.

pair = boolean -- (only for casmethod=Variational2RDM) treat the active orbitals with a pair variational 2-RDM method corresponding to the doubly occupied configuration interaction approximation. Default is false.

populations = string -- atomic-orbital population analysis: "Mulliken" and "Mulliken/meta-Lowdin". Default is "Mulliken".

spin = nonnegint -- twice the total spin S (= 2S). Default is 0.

state = nonnegint -- (for casmethod=ActiveSpaceSCF or casmethod=ActiveSpaceCI) sets the electronic state to be computed.  Default is 0, which is the ground state.

symmetry = string/boolean -- is the Schoenflies symbol of the abelian point-group symmetry which can be one of the following:  D2h, C2h, C2v, D2, Cs, Ci, C2, C1. true (default) finds the appropriate symmetry while false does not use symmetry.

threshold = float -- can be assigned to a float in the interval [0,1] which makes the ACSE treat active orbitals with occupations greater than the value 1-threshold and less than the value threshold as inactive (external) orbitals.  The default is -1.0 where the ACSE treats all of the orbitals from the CAS calculation as active.

unit = string -- "Angstrom" or "Bohr". Default is "Angstrom".

verbose = posint -- positive integer between 1 and 5 that controls printing. Default is 1.

Optional keywords from ActiveSpaceSCF, ActiveSpaceCI, or the Variational2RDM can also be used.

D. A. Mazziotti, Phys. Rev. Lett. 97, 143002 (2006). "Anti-Hermitian contracted Schrödinger equation: Direct determination of the two-Electron reduced density matrices of many-electron molecules"

D. A. Mazziotti, Phys. Rev. A 76, 052502 (2007). "Multireference many-electron correlation energies from two-electron reduced density matrices computed by solving the anti-Hermitian contracted Schrödinger equation"

J. W. Snyder Jr. and D. A. Mazziotti, J. Chem. Phys. 135, 024107 (2011). "Photoexcited conversion of gauche-1,3-butadiene to bicyclobutane via a conical intersection: Energies and reduced density matrices from the anti-Hermitian contracted Schrödinger equation"

S. E. Smart and D. A. Mazziotti, Phys. Rev. Lett. 126, 070504 (2021). "Quantum solver of contracted eigenvalue equations for scalable molecular simulations on quantum computing devices"

$\mathrm{with}\left(\mathrm{QuantumChemistry}\right)\:$

$\mathrm{molecule} ≔ \left[\left[''H''\,0\,0\,0\right]\,\left[''F''\,0\,0\,0.95\right]\right]\;$

$\mathrm{molecule}≔\left[\left[''H''\,0\,0\,0\right]\,\left[''F''\,0\,0\,0.95000000\right]\right]$

$\mathrm{output\_hf} ≔ \mathrm{ContractedSchrodinger}\left(\mathrm{molecule}\, \mathrm{basis}\=''dz''\right)\;$

$\mathrm{excited\_hf} ≔ \mathrm{ContractedSchrodinger}\left(\mathrm{molecule}\, \mathrm{basis}\=''dz''\,\mathrm{state}\=1\right)\;$

$\mathrm{mol} ≔ \mathrm{MolecularGeometry}\left(''Cys-Ser''\right)\;$

$\mathrm{mol}≔\left[\left[''S''\,\mathrm{-2.29090000}\,2.10530000\,\mathrm{-1.07690000}\right]\,\left[''O''\,2.52680000\,\mathrm{-1.19180000}\,\mathrm{-1.63380000}\right]\,\left[''O''\,\mathrm{-1.06460000}\,\mathrm{-0.04400000}\,1.69560000\right]\,\left[''O''\,2.66940000\,1.27060000\,1.34030000\right]\,\left[''O''\,1.68720000\,1.60900000\,\mathrm{-0.67760000}\right]\,\left[''N''\,0.14240000\,\mathrm{-0.64180000}\,\mathrm{-0.19310000}\right]\,\left[''N''\,\mathrm{-3.00870000}\,\mathrm{-1.66870000}\,0.29030000\right]\,\left[''C''\,1.44960000\,\mathrm{-0.55430000}\,0.41400000\right]\,\left[''C''\,\mathrm{-2.27570000}\,\mathrm{-0.56710000}\,\mathrm{-0.33790000}\right]\,\left[''C''\,\mathrm{-1.02220000}\,\mathrm{-0.37450000}\,0.51210000\right]\,\left[''C''\,2.44060000\,\mathrm{-1.50010000}\,\mathrm{-0.24810000}\right]\,\left[''C''\,\mathrm{-3.16790000}\,0.67650000\,\mathrm{-0.36510000}\right]\,\left[''C''\,1.91400000\,0.88100000\,0.28020000\right]\,\left[''H''\,1.35260000\,\mathrm{-0.79450000}\,1.47910000\right]\,\left[''H''\,\mathrm{-2.01660000}\,\mathrm{-0.86110000}\,\mathrm{-1.36160000}\right]\,\left[''H''\,0.09180000\,\mathrm{-0.84710000}\,\mathrm{-1.18670000}\right]\,\left[''H''\,2.11130000\,\mathrm{-2.53950000}\,\mathrm{-0.14810000}\right]\,\left[''H''\,3.43800000\,\mathrm{-1.39630000}\,0.19140000\right]\,\left[''H''\,\mathrm{-3.48660000}\,0.95790000\,0.64500000\right]\,\left[''H''\,\mathrm{-4.06810000}\,0.49580000\,\mathrm{-0.96240000}\right]\,\left[''H''\,\mathrm{-2.46140000}\,\mathrm{-2.52630000}\,0.22140000\right]\,\left[''H''\,\mathrm{-3.87150000}\,\mathrm{-1.84260000}\,\mathrm{-0.22420000}\right]\,\left[''H''\,3.16370000\,\mathrm{-1.81370000}\,\mathrm{-2.02500000}\right]\,\left[''H''\,\mathrm{-3.30210000}\,2.97810000\,\mathrm{-0.95790000}\right]\,\left[''H''\,2.98800000\,2.19320000\,1.24240000\right]\right]$

$\mathrm{data\_peptide} ≔\mathrm{ContractedSchrodinger}\left(\mathrm{mol}\, \mathrm{active\_cse}\=\left[10\,10\right]\right)\;$