8 Diffusion Monte Carlo Simulations for Phenol-Water Comple
8.3 Diffusion Monte Carlo simulation
Diffusion Monte Carlo (DMC) is a way of solving the Schrôdinger equation for the ground state of a multidimensional system. DMC solves the time-dependent Schrôdinger equation, by assuming that the multidimensional potential energy surface V of an electronic state is known. For a system of N
atoms of mass n ij, the time-dependent Schrôdinger equation is in atomic units,
( « . ') + ''( « ) '* ' (« •') (8-1)
where Y (gr,r) is the time-dependent wavefunction and q is the vector of the coordinates. By transforming the time frame into an imaginary one ( t = i t ), we obtain
(8-2) This is equivalent to the classic time-dependent diffusion process:
^ ' ^ O j V ^ C { q , t ) - k C ( g . t ) (8.3)
O t ;=|
where the diffusion constant is
The random walk technique is a way to reproduce a diffusional behaviour [212]. Hence, the Schrôdinger equation can be simulated by both a random walk for the kinetic energy term and a continuous weight assessment for the potential energy term. This can be done 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 represents a part of the wavefunction. To avoid the generation of replicas in the high-energy region of weak importance in the description of the wavefunction, a minimum threshold is set up [213] (See Figure 8.2).
New Models for Intermolecular Repulsion and their Application to van der W aals Helen H.Y. Tsui
8 Diffusion Monte Carlo Simulations for Phenol-Water Complex 148 Energy R eplica d eleted
0 0 0 0 0 0 0 0
Minimum thresholdy r
R eplacing itself R eplacing d eleted replicaFigure 8.2 Schematic diagram of population of replicas.
Every replica with a w eight sm aller than this lim it is im m ediately deleted. In this case, the replica with the largest w eight gives birth to tw o identical replicas each having h alf o f the parent's w eight; the first descendent replaces its father, the other substitutes the previously deleted replica. The final stage o f each tim e step involves updating the reference energy [210] (See F igure 8.3). T h e num ber o f replicas rem ains the sam e, and the final distribution o f replicas is used for calculating the vibrationally averaged energy and structure, and one-particle density.
N e w M odels for Interm olecular Repulsion and their Application to van d er W a a ls C o m p lexes and Crystals of O rg anic M olecules
H e le n H .Y . Tsui A ugust 2001
8 Diffusion Monte Carlo Simulations for Phenol-Water Complex 149
Time-dependent Schrôdinger equation in imaginary time => Isomorphous with diffusion equation with first-order rate term
a c d r y=i kC d y / ( r , T ) _ d z
Ê
( r . r )[v
( r . r ) Population of replicasSimulated by diffusion process, solved by random walk
Assigning weights to repiicas to simulate potential energy
i
Until convergence Final distribution of replicas =» Vibrationally averaged energy and geometryone-particle density
Figure 8.3 Flow chart of the core operations in the DMC algorithm. Each loop is a group of 100 RB-DMC time steps of length At .
R ep ro d u ced w ith p e rm issio n from van M ou rik [214].
8.3.1 Rigid-body diffusion Monte Carlo algorithm
The algorithm used in the DMC code is the rigid-body diffusion Monte Carlo (RB-DMC), first introduced by Buch [215]. This allows the simulation to be performed using a fixed geometry for each monomer in the cluster, in order to remove the intramolecular degrees of freedom. In the original publication on rigid-body diffusion Monte Carlo, Buch did not provide any details as to how the RB- DMC can be implemented. Therefore Benoit has devised his own RB-DMC algorithm in his DMC code (See Figure 8.4).
N e w M odels for Interm olecular R epulsion a n d their Application to v a n d er W a a ls Co m p lex es and Crystals of O rg anic M olecules
H e le n H .Y . Tsui August 2001
8 Diffusion Monte Carlo Simulations for Phenol-Water Complex 150 Start Initialisation of the replicas Initialisation of RB-DMC procedure No ► Equilibration No Y e s R esu lts
1
End C o n v erg en ce reach ed ? R andom G a u ssia n d isp la cem en tI
Energy evaluation for e a c h replica W eight calculation --- ^ RB-DMC sim ulation w C o n v erg en ce -
Figure 8.4 Flow chart of Rigid-Body Diffusion Monte Carlo (RB-DMC) program, Xdmc.
R ep ro d u ced w ith p erm issio n fro m B en o it [25].
8.3.2 Implementation
It is essential to understand the concepts o f how the algorithm can be im plem ented to provide an idea o f the processes within a DM C calculation. In depth details on the relevant theory can be found in B enoit's thesis [25], and F igure 8.4 gives the flow chart o f the procedures required for a DM C sim ulation. First o f all, a model potential is required for the phenol-w ater system s being investigated. O ur OPLS, FIT (as shown in C hapter 5) and non-em pirical m odel potentials {6311 gss_rpcm n model from C hapter 7) will be used. In order to ensure good overlap, it is best to start at the m inim um in the
N e w Mcxjels for Interm olecular Repulsion and their Application to v an d e r W a a ls C o m p lex es and C rystals ot O rg anic M olecules
H e len H .Y . Tsui August 2001
8 Diffusion Monte Carlo Simulations for Phenol-Water Complex_________________________ 151 potential; therefore the minima determined from ORIENT [132] are used as starting positions. This should also minimise the time that is required for the calculation.
8.3.3 Specifications
The equilibration is performed by diffusing 1000 replicas using time steps of 50 a.u.. This is carried out until the standard error associated with the reference energy is smaller than 0.25 %. The resulting ensemble of replicas is propagated, using time steps of 15 a.u., until the reference energy is converged to a standard of (XI % for the phenol-water system. This specification is used for all DMC simulations in this chapter.