The distributed multipole model

We have already discussed the deficiency of using the point charge model, in terms of approximating the atoms in the molecule as spherical. Therefore an improved method is the distributed multipole model, where it is essentially an extension and combination of atomic point charges with the central multipole expansion. Each atom in a molecule is approximately spherical; therefore it is possible to use the multipole expansion for each atom to model the charge distribution. The use of multipoles would ensure that the non-spherical features of the electron density are well represented and the distribution of multipoles on sites around the molecule should provide a better model for the overall molecular shape, when compared with the central expansion. There have been many different methods suggested, but in this thesis. Stone’s Distributed Multipole Analysis (DMA) [1,2] is used.

Stone’s DMA starts from the one-electron density matrix of any ab initio wavefunction expressed as a set of Gaussian functions. Then it uses the properties of these Gaussian orbitals to calculate efficiently the multiple moments at a number of sites,

= (2.18)

ij ij u V

Mfj is the element of the density matrix, and 0, ^ ( r ) is a normalised basis function,

expressed as a linear combination of Gaussian primitive functions T]^, which takes the form:

Vu = ( y - y j " ( z - z, r exp (r ] (2,19) for a function centred at and +jli^ +v^ equal to the angular momentum quantum number I, a.^ is the product of a normalisation constant and a contraction coefficient. Essentially, the method takes the product of two primitive functions that defines a new Gaussian function. Hence the charges' densities overlap at a centre P (Equation (2,20)) and a series of multipoles at that centre are obtained.

New Models for Intermolecular Repulsion and their Application to van der W aals Helen H.Y. Tsui

2 The Theory of Intermolecular Forces______________________________________ 31

p ^ ocA+p^ (2.20)

a + j8

F o r e x a m p l e , th e o v e r la p d e n s ity o f a n s f u n c tio n T]{s\ A ) a t A a n d a f u n c ti o n a t B is: 7) (^ ; A ) t ) ( p , ; B ) = e x p [ - a ( r - A )^ ] ( z - Zg ) e x p [ - ) 3 ( r - B ) ^ J = ( z - Z g ) e x p { - [ a ) 3 / ( a + )3 ) ] ( A - B ) " } e x p [ - ( a + )3 ) ( r - P f ] (2 .2 1 ) = { z - Z p + Z p - Z g ) e x p { - [ a / 3/ ( a + )3 ) ] ( A - B ) " } e x p [ - ( a + ^ ) ( r - B ) " ] T h e n , b y u s in g th is f o r m u la :

(^ ) = Ê È

it is possible to get from the expansion at a point P , new expansions centred at a new site S . In the equation, q and m refer to the components of the multipole moment and k and I to its rank. It is worth pointing out that the series (S ) is infinite but local moments of rank higher than quadrupole are expected to be small. This is due to the fact that the contribution from orbitals higher than j or p is normally very small, so the quadrupole (as a result of two p functions) should be the highest moment of significant magnitude. However, if |5 - P | is large. Le. we move the multipoles very far from the centre P, then higher order moments will still be large, as the expression for Qg„{^S) depends on |5 - *. S is usually taken as the nearest nuclear site in the expansion. The choice of sites 5 being arbitrary, we can choose those sites that are more suitable each time. The electrostatic potential due to a multipole series at a site S will be:

v ( r ) = 2 k - s r ' " ' Q . ( r - S ) Q ^ ( * ) (2 -2 3 )

Im

Now by comparison with the previous equation one can see that \r- 5 | should be much greater than

[ S - P ) , if convergence of the series is to be achieved. In other words, the points at which the potential is to be evaluated must all be further away from each site S than any of the centres P, which contribute charge density to S . This means that when choosing the sites, one should take care not to shift any multipoles very far from their original centre P . Choosing for example the nuclei as the DMA sites satisfies the previous criterion, since nuclei carry charge that needs to be included in the expansion, and in most ab initio calculations basis functions are centred on the nuclei anyway. The centre of a bond would also be a sensible choice as an additional DMA site, because it is situated along the line of accumulated charge density for high accuracy.

Distributed multipole analysis is a computationally efficient method of calculating electrostatic interactions, without using any approximations. This has the additional advantage of fast convergence that characterises the distributed multipole expansion. DMA is also valid at distances that are of importance in organic systems, such as the hydrogen bonds, whereas the conventional expansions are only valid at longer distances. An exceptional advantage of DMA is probably the fact that it is consistent with the ideas of bonding in molecules. For example, if a dipole (lone pair) is present, we

New Models for Intermolecular Repulsion and their Application to van der W aals Helen H.Y. Tsui

2 The Theory of Intermolecular Forces 32 expect to have it represented by a dipole, or a quadrupole to represent 7i-bonding electrons. (See Figure 2.2 for the tw o-dim ensional representations)

+z

•30

Figure 2.2 Diagrams representing the charge distributions that could be m odelled by a pure point charge, dipole, quadrupole and octadecapoles.

T he D M A m ultipoles are due to the distortion o f the charge distribution that occurs on bonding, since in general, the m om ents o f neutral spherical atom s are zero. D M A is a m ajor advance for m odelling the interm olecular forces, as the atom ic point charge m odel contradicts the theory o f bonding, and the central m ultipole expansion suffers from validity and convergence problem s. A com parison o f D M A with the atom ic point charge model will be discussed in C hapter 5.

T he D M A model has been used successfully in the study o f electrostatic interactions betw een m olecules, as well as the description o f the charge densities on individual m olecules. This has been tested on a range o f crystal structures by C oom bes et al. [70]. The electrostatic force in van der W aals hydrogen-bonded dim ers is dom inant in determ ining the strength and orientation dependence o f their interaction [60] as discussed below. This suggests that the total interaction may be approxim ated by a good representation o f the electrostatic forces, even w ithout an accurate description o f all the rem aining contributions.