The diffusion Monte Carlo (DMC) is a numerical method used to solve the Schrodinger equation by using Monte Carlo sampling. It was first used to perform electronic structure calculations by Anderson[1] before being applied to nuclear motions by Coker and Watts[2] . This quantum treatment is widely used to gain insights in the understanding of Van der Walls complexes because it provides the exact energy of the vibrational ground state. In fact, the high anharmonicity of the interactions which govern these complexes require the zero point energy to be included. The DMC algorithm, although easy to implement, has not the popularity it desserves because of its difficulties to get the excited states. Nevertheless, a good knowledge of the dynamics of the ground vibrational state is of real interest since it can give many important informations about the investigated system: dissociation energy, vibrationally averaged structures, rotational constants and wavefunctions. The accuracy of the results depends only on the quality of the potential used.
Assuming the multidimensional potential energy surface V of an electronic state is known, DMC solve the time dependent Schrodinger equation for a system of N atoms of mass m j . By transforming in this equation the time frame into an imaginary one, we get the expression of a classic time dependent diffusion process with a sink or source term.
The random walk technique, pioneered by Metropolis and Ulam[3] , is a generally accepted method to reproduce a diffusional behavior [4]. As a consequence, the Schrodinger equation can be simulated by both a random walk for the kinetic energy part and a continuous weight assessment for the potential energy term. This is achieved by generating a population of M replicas and by adjusting their weights according to their energy as they move randomly on the potential energy surface. Each of these "wavefunction particles" describes one possible geometry of the system and also represents (with more or less importance depending on its weight) a part of the wavefunction.
The random walk is implemented by moving at each time step all the atoms in the three directions x,y,z. By using the Stirling's approximation and by assuming x (or y or z) being not too far from the origin, the probability of being at a position x is given by a gaussian distribution centred at zero with a standard deviation taking into account the time of displacement and the mass mj of the shifted atom j .This deviation is calculated from the expression of the diffusion constant and the Einstein Smoluchowski equation [4,5].
The step following the diffusion is the calculation of the potential energy Vi of the replica i and the adjustment of its weight Wi. The major problem of this method arises from the non negligeable eventuality of generating a majority of replicas in the high energy region of weak importance in the description of the wavefunction. In fact, this would lead to a poor and unaccurate modelling of this latter. This is avoided by setting up a minimum threshold (usually 1/M) [6].
To prevent the wavefunction from its decay to zero, we introduce a non-zero reference energy Vref which will "trap" the lowest energy: this is the main trick of the diffusion Monte Carlo. This reference energy is updating at each time step over the simulation [1]. The exact ground state energy will be obtained by averaging this reference energy over the simulation.
Our simulation are intentionally performed by fixing the geometry of each monomer to remove the intramolecular high frequency vibrations. As a matter of fact, the only treatment of the intermolecular modes of larger time scale increases the efficiency of the simulation by permitting larger time scale to be used . This concept, namely the rigid body diffusion Monte Carlo simulation (RBDMC) was first introduced by Buch [7]. Rotational constants and the vibrationally averaged values of the internal coordinates of the minimum energy structure have been calculated by using the descendant weighting method [8].
()M. Quack and M.A. Suhm, J. Chem. Phys. 95,28 (1991)
[1] J.B. Anderson, J. Chem. Phys. 63,1499 (1975).
[2] D.F. Coker and R.O. Watts, Mol. Phys. 58,1113 (1986).
[3] N. Metropolis and S. Ulam, J.Am. Stat. Assoc. 44,335 (1949).
[4] A. Einstein, Ann. d. Phys. 17,549 (1905).
[5] A. Einstein, Ann. d. Phys. 19,371 (1906).
[6] M.A. Suhm and R.O.Watts, Phys. Rev. 204,293 (1991).
[7] V. Buch, J. Chem. Phys. 97,726 (1992).
[8] M.H. Kalos, Phys. Rev. A. 2,250 (1970).