/*! \page Pfaff_wfDoc Pfaffian Keyword: PFAFFIAN \section description Description A pfaffian (generalized determinant) or several pfaffians.
\f[ \Psi=\sum_i w_i \Psi_i \f] \f[ \Psi_i={\rm pf}\left[ \begin{array}{ccc} {\boldsymbol \xi}^{\uparrow\uparrow}_i & {\boldsymbol \Phi}^{\uparrow\downarrow}_i & {\boldsymbol\varphi}^{\uparrow}_i \\ -{\boldsymbol \Phi}^{\uparrow\downarrow T}_i & {\boldsymbol \xi}^{\downarrow\downarrow}_i & {\boldsymbol \varphi}^{\downarrow}_i \\ -{\boldsymbol\varphi}^{\uparrow T}_i & -{\boldsymbol\varphi}^{\downarrow T}_i & 0 \;\; \end{array} \right] \f] where submatrices \f$ \boldsymbol \xi \f$, \f$ \boldsymbol \Phi \f$ and \f$ \boldsymbol \varphi \f$ are defined in \ref Pfaffian_groupDoc section and ordered according the ORBITAL_ORDER section. \section Example Pfaffian wave function for the ground state of carbon pseudo-atom :
PFAFFIAN
  NPAIRS {  3  1  0  }
  PFWT {  1  }
  ORBITAL_ORDER {
  1    1    1
  1    1
  1
  }
  PAIRING_ORBITAL {
    ORBITALS_IN_PAIRING {  1  2  3  4  5  }
    OPTIMIZE_PF {
      NOCCUPIED 3
      SINGLET_ALL
    }
    TRIPLET_UU_COEF {
    0 0 0 0
    1 0 0
    0 0
    0
    }
    TRIPLET_DD_COEF {
    0 0 0 0
    0 0 0
    0 0
    0
    }
    SINGLET_COEF {
    1 0 0 0 0
    0 0 0 0
    0 0 0
    0 0
    0
    }
  }
  ORBITALS {
    CUTOFF_MO 
    MAGNIFY 1 
    INCLUDE c.basis 
    NMO 5 
    ORBFILE c.orb 
    CENTERS { USEATOMS }
    }
  }
\section options Options \subsection reqopt Required
Option Type Description
NPAIRS Section number of spin-up, spin-down and unpaired electrons (sum of these 3 numbers defines size of pfaffian matrix of size \f$ N \f$)
PFWT Section list of the weights of the pfaffians (defines also have many are there)
PAIRING_ORBITAL Section Section for \subpage Pfaffian_groupDoc
ORBITALS Section input for a \ref MO_matrixDoc
\subsection optopt Optional
Option Type Default Description
ORBITAL_ORDER Section i-th pairing orbital for i-th pfaffian matrix Contains upper-diagonal of \f$ N \times N \f$ matrix of pairing occupation numbers
OPTIMIZE_PFWT keyword off Optimize the pfaffian weights.
*/ /*! \page Pfaffian_groupDoc Pairing Orbital Keyword: PAIRING ORBITAL \section description Description Defines the coefficients of linear combinations of one-particle orbitals from which the pairing orbitals are constructed. The normalization is everywhere such that sum of squares of independent coefficients is equal to 1. \section options Options \subsection reqopt Required
Option Type Default Description
TRIPLET_UU_COEF Section All zero \f$ N(N-1)/2 \f$ normalized coefficients for triplet spin-up spin-up pair of form: \f$ \chi(1,2)= {1 \over \sqrt(2)} \sum_{i>j} a_{i,j}( \varphi_i(1)\varphi_j(2)- \varphi_j(1)\varphi_i(2))\f$
TRIPLET_DD_COEF Section All zero same as above for spin-down case
SINGLET_COEF Section 1 on diagonal for all occupied orbitals, zero otherwise \f$ N(N+1)/2 \f$ normalized coefficients for singlet spin-up spin-down pair of form: \f$ \phi(1,2)= \sum_{i} s_{i,i} \varphi_i(1)\varphi_j(2) + {1 \over \sqrt(2)} \sum_{i>j} s_{i,j}( \varphi_i(1)\varphi_j(2)+\varphi_j(1)\varphi_i(2))\f$
UNPAIRED_COEF Section All zero # of spin down \f$\times\f$ # normalized coefficients for unpaired electron orbital of form: \f$ \varphi'_{i}(1)= \sum_{j} c_{i,j}\varphi_j(1)\f$
\subsection optopt Optional
Option Type Default Description
ORBITALS_IN_PAIRING Section 1 to NMO list of orbitals involved in pairing
OPTIMIZE_PF Section empty bellow are keywords used for determination of which pairing coefficients to be optimized
KEYWORDDescription
TRIPLET_UU_DIAG First off-diagonal triplet up up coefficients
TRIPLET_UU_ALL All triplet up up coefficients
TRIPLET_DD_DIAG First off-diagonal triplet down down coefficients
TRIPLET_DD_ALL All triplet down down coefficients
SINGLET_DIAG First off-diagonal triplet up up coefficients
SINGLET_ALL All singlet coefficients
NOCCUPIED number of occupied orbs used with keyword VIRTUAL (default = number of up electrons)
TRIPLET_UU_VIRTUAL same as above just for virtual orbitals
TRIPLET_UU_VIRTUAL_DIAG same as above just for virtual orbitals
TRIPLET_DD_VIRTUAL same as above just for virtual orbitals
TRIPLET_DD_VIRTUAL_DIAG same as above just for virtual orbitals
SINGLET_VIRTUAL same as above just for virtual orbitals
SINGLET_VIRTUAL_DIAG same as above just for virtual orbitals
*/