Excitation
Contents: Table of Contents  energies  Chapter 6  Frequency Dependent Polarizabilities

Dispersion Coefficients

 

• Higher order dispersion coefficients

The dispersion energy between two systems is expessed as a power series
 
  $$E_{disp} = - \sum_n {C_n \over R^n}$$  

The above expression is for spherical systems - for general molecules there are angular factors as well. The C_n coefficients can be related to integrals of the polarizability at imaginary frequencies. The program calculates the integrals
 
  $$X_{ijkl} = {1\over {2\pi}} \int\alpha^A_{ij}(i\omega)\alpha^B_{kl}(i\omega)d\omega$$  

where $\alpha^A$ and $\alpha^B$ are the polarizabilities of systems A and B

The isotropic (spherically averaged) C₆ coefficient is
 
  $$C_6 = {3 \over \pi} \int {\bar\alpha^A}(i\omega) {\bar\alpha^B}(i\omega)d\omega$$  

where $\bar\alpha$ is the average polarizability,
 
  $$\bar\alpha = {1\over 3}(\alpha_{xx}+\alpha_{yy}+\alpha_{zz})$$  

Here is an example of a dataset for a calculation on argon.

TITLE
ARGON dispersion (C6) coefficient        
BASIS 631GE  
ATOMS
ARGON   18.0   0   0   0
END
NOPRINT OCCVECTORS
DISPERSION C6
START
 
FINISH

The output consists of the values of the integrals X_ijkl defined above. Since this is a spherical system all the integrals are equal in this case. The isotropic C₆ coefficient for the argon-argon interaction is also given. The output for the argon calculation looks like this,

Dispersion energy integrals from RPA solution

 Terms contributing to C 6
    (X  ,X  )    (X  ,X  )         8.58202
    (Y  ,Y  )    (X  ,X  )         8.58202
    (Y  ,Y  )    (Y  ,Y  )         8.58202
    (Z  ,Z  )    (X  ,X  )         8.58202
    (Z  ,Z  )    (Y  ,Y  )         8.58202
    (Z  ,Z  )    (Z  ,Z  )         8.58202

 Isotropic C6 coefficient        51.49210  (atomic units)

As another example consider the nitrogen molecule,

TITLE
N2 dispersion coefficients
SYMMETRY
DNH 2
END

ANGSTROM
ATOMS
N 7 0 0 0.548    
LIBRARY N631GE
2 F 1
1 0.5 1.0
END
END

DISPERSION C6
START
FINISH

The output in this case is as follows.

Dispersion energy integrals from RPA solution

 Terms contributing to C 6
    (X  ,X  )    (X  ,X  )         8.39510
    (Y  ,Y  )    (X  ,X  )         8.39510
    (Y  ,Y  )    (Y  ,Y  )         8.39510
    (Z  ,Z  )    (X  ,X  )        12.45822
    (Z  ,Z  )    (Y  ,Y  )        12.45822
    (Z  ,Z  )    (Z  ,Z  )        18.77898

 Isotropic C6 coefficient        68.12817  (atomic units)

Note that the basic integrals are not all equal, so there would be angular factors in the dispersion interaction. Expressions for the angular factors for simple molecules (spherical tops) are given by Langhoff (J. Chem. Phys.,55 (1971) 2126). The angular terms may be assembled from the X_ijkl integrals.

The program will calculate dispersion energies for closed-shell SCF and DFT.

Higher order dispersion coefficients

The program can also calculate C₇ coefficients (these are zero for centrosymmetric systems). Just generalise the appropriate command to read,

DISPERSION C6 C7

The output is in the same format as that considered above, but slightly more complicated, for example this is from a calculation on hydrogen fluoride.

Dispersion energy integrals from RPA solution


 Terms contributing to C 6
    (X  ,X  )    (X  ,X  )         1.84694
    (Y  ,Y  )    (X  ,X  )         1.84694
    (Y  ,Y  )    (Y  ,Y  )         1.84694
    (Z  ,Z  )    (X  ,X  )         2.49476
    (Z  ,Z  )    (Y  ,Y  )         2.49476
    (Z  ,Z  )    (Z  ,Z  )         3.41510

 Terms contributing to C 7
    (XX ,Z  )    (X  ,X  )        -3.30012
    (XX ,Z  )    (Y  ,Y  )        -3.30012
    (XX ,Z  )    (Z  ,Z  )        -4.48277
    (YY ,Z  )    (X  ,X  )        -3.30012
    (YY ,Z  )    (Y  ,Y  )        -3.30012
    (YY ,Z  )    (Z  ,Z  )        -4.48277
    (ZZ ,Z  )    (X  ,X  )         6.60024
    (ZZ ,Z  )    (Y  ,Y  )         6.60024
    (ZZ ,Z  )    (Z  ,Z  )         8.96555
    (XZ ,X  )    (X  ,X  )         4.21575
    (XZ ,X  )    (Y  ,Y  )         4.21575
    (XZ ,X  )    (Z  ,Z  )         5.68169
    (YZ ,Y  )    (X  ,X  )         4.21575
    (YZ ,Y  )    (Y  ,Y  )         4.21575
    (YZ ,Y  )    (Z  ,Z  )         5.68169


 Isotropic C6 coefficient        13.85462

The main difference is that the C₇ coefficients are constructed from elements involving the dipole-quadrupole polarizability as well as the dipole-dipole polarizability, e.g. the term labelled (xx,z) (x,x) is the integral,
 
  $${1\over{2\pi}}\int A_{xx,z}(i\omega)\alpha_{xx}(i\omega)d\omega$$  

and so on with the other terms.