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6.13 One-electron operators and expectation values (GEXPEC)

The operators for which expectation values are requested, are specified by keywords on the global GEXPEC directive. By default, only dipole moments are computed. The first letter G is optional, but should be used to avoid confusion with program specific EXPEC cards, which have the same form as GEXPEC. For all operators specified on the GEXPEC card, expectation values are computed in all subsequent programs that generate the first-order density matix. This is always the case for variational wavefunctions, i.e., HF, DFT, MCSCF, MRCI. For non-variational wavefunctions such as MP2, QCISD, QCISD(T), CCSD, or CCSD(T) the density matix is not computed by default, since this requires considerable additional effort (solving z-vector equations). The GEXPEC directive does not affect such programs. In some cases [currently for MP2, QCISD, QCISD(T)] the EXPEC directive that is specific to those programs can be used to request the property calculation.

For a number of operators it is possible to use generic operator names, e.g., DM for dipole moments, which means that all three components DMX, DMY, and DMZ are computed. Alternatively, individual components may be requested.

The general format is as follows:

[G]EXPEC,opname[,][icen,[x,y,z]],...

where

opname
operator name (string), either generic or component.
icen
z-matrix row number or z-matrix symbol used to determine the origin (x,y,z must not be specified).
If icen$=0$ or blank, the origin must be specified in x,y,z

Several GEXPEC cards may follow each other, or several operators may be specified on one card.

Examples:

GEXPEC,QM computes quadrupole moments with origin at (0,0,0),

GEXPEC,QM1 computes quadrupole moments with origin at centre 1.

GEXPEC,QM,O1 computes quadrupole moments with origin at atom O1.

GEXPEC,QM,,1,2,3 computes quadrupole moments with origin at (1,2,3).

The following table summarizes all available operators:


Table 5: One-electron operators and their components
Generic Parity Components Description
name      
OV 1   Overlap
EKIN 1   Kinetic energy
POT 1   potential energy
DELTA 1   delta function
DEL4 1   $\Delta^4$
DARW 1   one-electron Darwin term,
      i.e., DELTA with appropriate factors
      summed over atoms.
MASSV 1   mass-velocity term,
      i.e., DEL4 with appropriate factor.
REL 1   total Cowan-Griffin Relativistic correction,
      i.e., DARW+MASSV.
DM 1 DMX, DMY, DMZ dipole moments
SM 1 XX, YY, ZZ, XY, XZ, YZ second moments
TM 1 XXX, XXY, XXZ, XYY, XYZ,  
    XZZ, YYY, YYZ, YZZ, ZZZ third moments
MLTPn 1 all unique Cartesian products of order $n$ multipole moments
QM 1 QMXX, QMYY, QMZZ, QMXY, QMXZ, QMYZ, quadrupole moments and $R^2$
    QMRR=XX + YY + ZZ,  
    QMXX=(3 XX - RR)/2,  
    QMXY=3 XY / 2 etc.  
EF 1 EFX, EFY, EFZ electric field
FG 1 FGXX, FGYY, FGZZ, FGXY, FGXZ, FGYZ electric field gradients
DMS 1 DMSXX, DMSYX, DMSZX,  
    DMSXY, DMSYY, DMSZY,  
    DMSXZ, DMSYZ, DMSZZ diamagnetic shielding tensor
LOP -1 LX, LY, LZ Angular momentum operators $\hat L_x$, $\hat L_y$, $\hat L_z$
LOP2 1 LXLX, LYLY, LZLZ, one electron parts of products of
    LXLY, LXLZ, LYLZ angular momentum operators.
    The symmetric combinations $\frac{1}{2} (\hat L_x \hat L_y+\hat L_y \hat L_x)$ etc. are computed
VELO -1 D/DX, D/DY, D/DZ velocity
LS -1 LSX, LSY, LSZ spin-orbit operators
ECPLS -1 ECPLSX, ECPLSY, ECPLSZ ECP spin-orbit operators

Expectation values are only nonzero for symmetric operators (parity=1). Other operators can be used to compute transition quantities (spin-orbit operators need a special treatment).



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