Molecular Properties and Conversion Factors
 
These two papers are of general interest:
 A.D.Buckingham, J.Chem.Phys. 30, 1580-1585(1959).
 D.Neumann, J.W.Moskowitz J.Chem.Phys. 49, 2056-2070(1968).
The first deals with multipoles, and the second with other 
properties such as electrostatic potentials.
 
All units are derived from the atomic units for distance 
and the monopole electric charge, as given below.
 
distance                 1 au = 5.291771E-09 cm
 
monopole                 1 au = 4.803242E-10 esu
                        1 esu = sqrt(g-cm**3)/sec
 
dipole                   1 au = 2.541766E-18 esu-cm
                      1 Debye = 1.0E-18 esu-cm
 
quadrupole               1 au = 1.345044E-26 esu-cm**2
                 1 Buckingham = 1.0E-26 esu-cm**2
 
octopole                 1 au = 7.117668E-35 esu-cm**3
 
electric potential       1 au = 9.076814E-02 esu/cm
 
electric field           1 au = 1.715270E+07 esu/cm**2
                  1 esu/cm**2 = 1 dyne/esu
 
electric field gradient  1 au = 3.241390E+15 esu/cm**3
 
The atomic unit for the total electron density is 
electron/bohr**3, but 1/bohr**3 for an orbital density.
 
The atomic unit for spin density is excess alpha spins per 
unit volume, h/4*pi*bohr**3.  Only the expectation value is 
computed, with no constants premultiplying it.
 
IR intensities are printed in Debye**2/amu-Angstrom**2. 
These can be converted into intensities as defined by 
Wilson, Decius, and Cross's equation 7.9.25, in km/mole, by 
multiplying by 42.255.  If you prefer 1/atm-cm**2, use a 
conversion factor of 171.65 instead.  A good reference for 
deciphering these units is A.Komornicki, R.L.Jaffe 
J.Chem.Phys. 1979, 71, 2150-2155.  A reference showing how 
IR intensities change with basis improvements at the HF 
level is Y.Yamaguchi, M.Frisch, J.Gaw, H.F.Schaefer, 
J.S.Binkley, J.Chem.Phys. 1986, 84, 2262-2278.  

Raman activities in A**4/amu multiply by 6.0220E-09 for
units of cm**4/g.  One of the many sources explaining how 
activity relates to intensity is D.Michalska, R.Wysokinski 
Chem.Phys.Lett. 403, 211-217(2005)


Polarizabilities

Static polarizabilities are named alpha, beta, and gamma; 
these are called the polarizability, hyperpolarizability, 
and second hyperpolarizability.  They are the 2nd, 3rd, and 
4th derivatives of the energy with respect to uniform 
applied electric fields, with the 1st derivative being the 
dipole moment.

It is worth mentioning that a uniform (static) electric 
field can be applied using $EFIELD, if you wish to develop 
custom usages, but $EFIELD input must not be given for any 
kind of run discussed below.

A general approach to computing static polarizabilities is 
numerical differentiation, namely RUNTYP=FFIELD, which 
should work for any energy method provided by GAMESS.  A 
sequence of computations with fields applied in the x, y, 
and/or z directions will generate the three alpha, beta, 
and gamma tensors.  See $FFCALC for details.  Analytic 
computation of all three tensors is available for closed 
shells only, see RUNTYP=TDHF and $TDHF input, or TDDFT=HPOL 
and $TDDFT input.  If you need to know just the static 
alpha polarizability, see POLAR in $CPHF during any 
analytic hessian job.

A break down of the static alpha polarizability in terms of 
contributions from individual localized orbitals can be 
obtained by setting POLDCM=.TRUE. in $LOCAL.  Calculation 
will be by analytic means, unless POLNUM in that group is 
selected.  This option is available only for SCFTYP=RHF.  
The keyword LOCHYP in $FFCALC gives a similar analysis for 
all three static polarizabilities, determined by numerical 
differentiation.

Polarizabilities in a static electric field differ from 
those in an oscillating field, such as a laser produces.  
These are called frequency dependent alpha, beta, or gamma, 
and in the limit of entering a zero frequency, become the 
static quantities discussed just above.

For RHF cases, various frequency dependent alpha, beta, and 
gamma polarizabilities can be generated, depending on the 
experiment.  A particularly easy one to understand is 
'second harmonic generation', governed by a beta tensor 
describing the absorption of two photons with the emission 
of one photon at doubled frequency.  See RUNTYP=TDHF, and 
papers listed under $TDHF, for many other non-linear 
optical experiments.  A program for the computation of the 
frequency dependent beta hyperpolarizability at the DFT 
level is also available, for closed shell molecules:  see 
TDDFT=HPOL and keywords in $TDDFT input.

Nuclear derivatives of the dipole moment and the various 
polarizabilities are also of interest.  For example, 
knowledge of the derivative of the dipole with respect to 
nuclear coordinates yields the IR intensity.  Similarly, 
the nuclear derivative of the static alpha polarizability 
gives Raman activities: see RUNTYP=RAMAN.  Analytically 
computed 1st or 2nd nuclear derivatives of static or 
frequency dependent polarizabilities are available for 
SCFTYP=RHF, see RUNTYP=TDHFX and $TDHFX, giving rise to 
experimental observations such as resonance Raman and 
hyper-Raman.

Finally, instead of considering polarizabilities to be a 
function of real frequencies, they can be considered to be 
dependent on the imaginary frequency.  The imaginary 
frequency dependent alpha polarizability can be computed 
analytically for SCFTYP=RHF only, using POLDYN=.TRUE. in 
$LOCAL.  Integration of this quantity over the imaginary 
frequency domain can be used to extract C6 dispersion 
constants.

Polarizabilities are tensor quantities.  There are a number 
of different ways to define them, and various formulae to 
extract "scalar" and "vector" quantites from the tensors.  
A good reference for learning how to compare the output of 
a theoretical program to experiment is
    A.Willetts, J.E.Rice, D.M.Burland, D.P.Shelton
    J.Chem.Phys. 97, 7590-7599(1992)