Van der Waals dispersion coefficients
The program DISPER computes Van der Waals dispersion coefficients up to C10 for two arbitrary closed-shell molecules. ADF itself can already compute some C6 and C8 coefficients between two identical closed-shell molecules. These coefficients describe the long-range dispersion interaction between two molecules. It requires previous ADF-TDDFT calculations for the polarizability tensors at imaginary frequencies for the two interacting molecules. Each such ADF calculation produces a file TENSOR (if suitable input for ADF is given). The TENSOR files must be renamed tensorA and tensorB, respectively and must be present as local files for DISPER. The DISPER program takes no other input and prints a list of dispersion coefficients.
A schematic example, taken from the set of sample runs, for the usage of DISPER is the following:
Step1: run ADF for, say, the HF molecule. In the input file you specify the RESPONSE data block:
RESPONSE
MaxWaals 8 ! Compute dispersion coefficients up to C8
ALLTENSOR ! This option must be specified in the ADF calc for a
! subsequent DISPER run
ALLCOMPONENTS ! Must also be specified for DISPER
End
At the end of the run, copy the local file 'TENSOR' to a file 'tensorA'. For simplicity, we will now compute the dispersion coefficients between two HF molecules. Therefore, copy 'tensorA' to 'tensorB'.
Now run DISPER (without any other input). It will look for the local files 'tensorA' and 'tensorB' and compute corresponding dispersion coefficients to print them on standard output.
The output might look something like this:
DISPER 2000.02 RunTime: Apr04-2001 14:14:13 ********** C-COEFFICIENTS ********** n LA KA LB KB L coefficient(Y) coefficient(P) 6 0 0 0 0 0 28.29432373 28.29432373 6 2 0 0 0 2 7.487547697 3.348533127 8 0 0 0 0 0 416.1888455 416.1888455 8 0 0 2 0 2 0.4323024202E-05 0.1933315197E-05 8 2 0 0 0 2 402.3556946 179.9389368 8 2 0 2 0 4 0.4238960180E-05 8 4 0 0 0 4 -36.67895539 -12.22631846 8 4 0 2 0 6 -0.2000286301E-05
The n-value in the first column refers to the long-range radial interaction. The case n=6 refers to the usual dipole-dipole type interaction related to a 1/R6 dependence in the dispersion energy. The n=7 case relates to a dipole-quadrupole polarizability on one system and a dipole-dipole polarizability on the other (this is not symmetric!). The n=8 term may contain contributions from a quadrupole-quadrupole polarizability on one system in combination with a dipole-dipole polarizability on the other as well as contributions from two dipole-quadrupole polarizabilities.
Terms which are zero by symmetry are not printed. In the example above, this is the case for all n=7 terms, because the systems (apparently) are too symmetric to have a nonzero dipole-quadrupole polarizability. The best known and most important coefficients are the isotropic ones, determining the purely radial dependence of the dispersion energy. They are characterized by the quantum numbers: 6 0 0 0 0 0 (or 8 0 0 0 0 0 etc.) Other combinations of quantum numbers refer to different types of angular dependence. The complete set determines the dispersion energy for arbitrary orientations between the two subsystems A and B.
The complete expressions are rather involved and lengthy. We refer the interested reader to the paper [13] which contains a complete description of the meaning of the various parts of the output, as well as references to the earlier literature which contain the mathematical derivations. In particular, a useful review, which was at the basis of the ADF implementation, is given in [14]. Of particular significance is Eq.(8) of the JCP paper mentioned above, as it defines the meaning of the calculated coefficients Cn(LA,KA,LB,KB,L) as printed above.
For highly symmetric systems, a different convention is sometimes employed. It is based on Legendre polynomials (hence the 'P' in the final column) instead of on the spherical harmonics (the 'Y' in the column before the last). The 'P' coefficients are defined only for those coefficients that are nonzero in highly symmetric systems and never contain additional information with respect to the 'Y' coefficients. They are defined [Eq. (14) in the mentioned J. Chem. Phys. paper] in terms of the 'Y' coefficients by:
CnL = (-1)LCnL,0,0,0,L/√(2L+1)
Because the quality of the dispersion coefficients is determined by the quality of the polarizabilities that are the input for DISPER, it is important to get good polarizabilities from ADF. For that it is important, in the case of small systems, to use an asymptotically correct XC potential (several choices are available in ADF, such as SAOP or GRAC) and a basis set containing diffuse functions. We refer to the ADF User's Guide for details.
