Contents: Table of Contents   Intermolecular Perturbation Theory  Chapter 6  Origin and Gauge Dependence

Distributed Polarizabilities

An earlier section in this chapter considered the calculation of distributed multipole moments by integration over the Voronoi polyhedra surrounding each atom. It is possible to obtain distributed polarizabilities by a similar technique. The procedure is first to carry out an analytic polarizability calculation (which means the method must be SCF, DFT or MP2) in order to obtain the density matrices as perturbed by the field, and then to use the NUMPOL command.

Here is an example dataset - MP2 polarizabilites for water molecule

TITLE
test of distributed properties                               
BASIS SADLEJ
SYMMETRY 
CNV 2
END
ATOMS
O 8 0.0 0.0 0.0   
H2 1 0.0 1.4305 1.1072
END
NOPRINT OCCVECTORS
ORIGIN COM
CONVERGE 8
MP2
POLARISABILITY
START
FINISH

If one does a distributed calculation of multipole moments then one gets local charges, dipoles etc. on each site

The total molecular dipole can be reconstructed from these by the formula
 
 

$$\mu = \sum_a \mu^a + q^a r^a$$  

where $\mu^a$ is the local (ie atomic) dipole and q^a is the atomic charge, which is located at position r^a It follows that a definition of the local polarizability is
 
 

$$\alpha^a = \sum_a {{\partial \mu^a} \over {\partial F}} + {{\partial q^a} \over {\partial F}}r^a$$  

This definition has been used by various people before, but it has a problem in that the local polarizabilities depend on the choice of origin as a consequence of the charge flow term.

However the program also contains a new definition of the local polarizability, which is origin independent - it does this by eliminating the charge flow term in teh above expression by allowing the boundaries of the atoms to change in such a way that their charges remain constant.

Both defintions of the polarizability are given. This first, in the example given, produces this,

 TOTAL ATOMIC DIPOLE POLARISABILITIES
 (original definition including derivatives of charges)
                     Fx        Fy        Fz
    O            6.50878   0.00000   0.00000
    O            0.00000   2.88964   0.00000
    O            0.00000   0.00000   4.77820

    H2           1.52933   0.00000   0.00000
    H2           0.00000   3.58311  -1.96748
    H2           0.00000  -1.67327   2.48764

    H2           1.52933   0.00000   0.00000
    H2           0.00000   3.58311   1.96748
    H2           0.00000   1.67327   2.48764

and the newer definition produces this,

 TOTAL ATOMIC DIPOLE POLARISABILITIES
 ( including relaxation of boundaries)
                     Fx        Fy        Fz
    O            6.50878   0.00000   0.00000
    O            0.00000   5.93901  -0.00075
    O            0.00000   0.00011   6.09238
 average atomic polarisability    6.18006
 fraction of molecular polarisability    0.63112

    H2           1.52933   0.00000   0.00000
    H2           0.00000   2.05831  -0.74870
    H2           0.00000  -0.56293   1.83055
 average atomic polarisability    1.80606
 fraction of molecular polarisability    0.18444

    H2           1.52933   0.00000   0.00000
    H2           0.00000   2.05893   0.74950
    H2           0.00000   0.56300   1.83058
 average atomic polarisability    1.80628
 fraction of molecular polarisability    0.18446

Distributed Polarizabilities (alternative)

The program also contains a second algorithm for distributed polarizabilities, due to Anthony Stone and Ruth LeSueur.  The following should work if your version of Cadpac is reasonably up to date:

MULTIPOLES
MATRICES
DELETE H
LIMIT 2
START

PUNCH POLARIZABILITIES

POLARIZABILITIES
SKIP SCF INTEGRALS (if you've already done the SCF)
PERTURBATIONS MULTIPOLAR
START

Various caveats:

(i) This can use a lot of memory. It calculates a matrix over the basis set for every multipole moment operator on every site.

(ii)  Currently there is a limit of  60 for the total number of multipole moment operators. For a large molecule this may not be enough. The limit can be increased, but to do it safely you need the latest version of the program, in which the relevant number is given as a parameter rather than being hard-coded.

(iii) On the other hand there is probably not much point in including H atoms or in taking the rank above 2 for the others.

(iv)  The "punch" output includes labels identifying the atoms and multipoles. They can be left in the file -- they are enclosed in parentheses, so Orient ignores them. The actual numbers are in the right order for Orient. It is helpful if you give different labels to each atom, otherwise you might not know which is which. Only the first 3 characters of the atom name are used in the label.

This is the beginning of a typical punch output file (though I wouldn't
recommend the basis for this purpose):

PHENOL (torsion C-C-O-H = 0 degrees) 631G**, distributed polarizabilities
C6           0.00000000     0.00000000     0.00000000
O            0.00000000    -2.55409903     0.00000000
C1           2.20940385     1.40475746     0.00000000
C2           2.08354249     4.02260633     0.00000000
C3          -0.22515601     5.24541699     0.00000000
C4          -2.42610521     3.81722626     0.00000000
C5          -2.32895544     1.20886548     0.00000000
Static polarizabilities
(     C6-Q00       C6-Q10       C6-Q11c      C6-Q11s      O-Q00        O-Q10  )
(     O-Q11c       O-Q11s       C1-Q00       C1-Q10       C1-Q11c      C1-Q11s)
(     C2-Q00       C2-Q10       C2-Q11c      C2-Q11s      C3-Q00       C3-Q10 )
(     C3-Q11c      C3-Q11s      C4-Q00       C4-Q10       C4-Q11c      C4-Q11s)
(     C5-Q00       C5-Q10       C5-Q11c      C5-Q11s             )
(C6-Q00  )
    1.96029619   0.00000000   0.01879017  -0.12203093  -0.45243640   0.00000000
   -0.00959402  -0.25142044  -0.64634756   0.00000000  -0.19050366   0.14954748
   -0.01467777   0.00000000   0.08620541   0.14906367  -0.24488039   0.00000000
    0.01040476  -0.21208428  -0.03227115   0.00000000  -0.15916710   0.11015726
   -0.56968293   0.00000000   0.28275956   0.15901509
(C6-Q10  )
    0.00000000   2.85796280   0.00000000   0.00000000   0.00000000   0.00510965
    0.00000000   0.00000000   0.00000000   0.07021326   0.00000000   0.00000000
    0.00000000  -0.23271481   0.00000000   0.00000000   0.00000000  -0.00236659
    0.00000000   0.00000000   0.00000000  -0.13430135   0.00000000   0.00000000
    0.00000000   0.06661940   0.00000000   0.00000000
(C6-Q11c )
    0.01879003   0.00000000   1.14359389   0.00021206  -0.00519867   0.00000000
   -0.26631727   0.07041919  -0.00304057   0.00000000  -0.07812799  -0.19112099
    0.00743470   0.00000000  -0.12911275  -0.00076117  -0.02352733   0.00000000
    0.11075521   0.01737413   0.06424262   0.00000000  -0.05451351  -0.02057198
   -0.05870077   0.00000000  -0.16918438   0.07771387
(C6-Q11s )
   -0.12203115   0.00000000   0.00021219   1.59410078   0.54678348   0.00000000
    0.08307574   0.22353262  -0.08718789   0.00000000  -0.15508679   0.01294980
   -0.18422220   0.00000000   0.25108440   0.35011362   0.02685400   0.00000000
    0.02808510  -0.09458787  -0.21416538   0.00000000  -0.36503492   0.22521253
    0.03396915   0.00000000   0.37492970   0.20092419
(O-Q00   )
   -0.45243636   0.00000000  -0.00519887   0.54678314   1.10639618   0.00000000
    0.25766708   0.03450442  -0.24364525   0.00000000  -0.20808845   0.00793657
   -0.03021500   0.00000000  -0.03649285   0.03623330  -0.13197213   0.00000000
    0.00389471  -0.11673244  -0.04304661   0.00000000   0.03887836   0.04774123
   -0.20508084   0.00000000   0.18068540   0.04578894
(O-Q10   )
    0.00000000   0.00510965   0.00000000   0.00000000   0.00000000   2.79529455
    0.00000000   0.00000000   0.00000000  -0.05739006   0.00000000   0.00000000
    0.00000000  -0.01284923   0.00000000   0.00000000   0.00000000  -0.00398782
    0.00000000   0.00000000   0.00000000  -0.03421678   0.00000000   0.00000000
    0.00000000  -0.04746352   0.00000000   0.00000000
etc.