*From*: jkl[ AT ]ccl.net*Subject*: Basis sets intro. Part 3/3*Date*: Mon, 04 Mar 91 17:55:02 EST

Polarization and diffuse functions ---------------------------------- The original contractions derived from atomic Hartree-Fock calculations are frequently augmented with other functions. The most popular are the polarization and diffuse functions. The polarization functions are simply functions having higher values of L than those present in occupied atomic orbitals for the corresponding atom. At least for me, there is some ambiguity here, since for lithium, the p-type functions are not considered polarization functions, while for sulphur, the d-functions are considered polarization functions. In both cases these orbitals are not populated in the ground electronic state of the atom. The reason for including p-type functions in the Li and Be atoms, even in the minimal basis sets, is prac- tical, however. Without these functions, the results are extremely poor. The reason for not including d-type functions for sulphur should be the same as for other atoms, i.e., you can obtain reasonable results without them. I wish, I could believe that. The exponents for polarization functions cannot be derived from Hartree-Fock calculations for the atom, since they are not populated. However, they can be estimated from correlated calculations involving atoms. In practice, however, these exponents are estimated "using well established rules of thumb or by explicit optimization" (Dunning, 1989). The polarization functions are important for reproducing chemical bonding. They were frequently derived from optimizing exponents for a set of molecules. They should also be included in all correlated calculations. They are usually added as uncontracted gaussians. It is important to remember that adding them is costly. Augmenting basis set with d type polarization functions adds 5 (or 6) basis function on each atom while adding f type functions adds 7 (or 10, if spurious combinations are not removed). This brings us to the problem of specifying the number of d, f, g, etc. polarization functions in a form of some compact notation. Unfortunately, there is no provision for this information in the notations described above. Pople's 6-31G* basis uses 6 d type functions as polarization functions, while 6-311G* uses 5 of them. The () notation is no better. If the paper does not say explicitly how many d or f functions are used, you are on your own. The only way to find out is to repeat the calcula- tions or contact the author. Many papers do not specify this important information. The Pople's group introduced yet another more general notation to encode type of polarization functions. The easiest way is to explain an example. The 6-31G** is synonymous to 6-31G(d,p); the 6-311G(3d2f,2p) represents 6-311G set augmented with 3 functions of type d and 2 functions of type f on heavy atoms, and 2 functions of type p on hydrogens or specifically (6311,311,111,11)/(311,11), i.e. (11s,4p,3d,2f/5s,2p) -> [4s3p3d2f/3s2p] contraction. The 6 d-type polarization function is added to 6-31G set, while only 5 to 6-311G. For both 6-31G and 6-311G set, f-type functions are added in groups of 7. Polarization functions are, as a rule, used uncontracted. More information can be found in the following papers: (Dunning, 1989), (Francl et al., 1982), (Gutowski et al., 1987), (Jankowski, 1985), (Krishnan et al., 1980). The basis sets are also frequently augmented with the so-called diffuse functions. The name says it all. These gaussians have very small exponents and decay slowly with distance from the nucleus. Diffuse gaussians are usually of s and p type, however sometimes diffuse polarization functions are also used. Diffuse functions are necessary for correct description of anions and weak bonds (e.g. hydrogen bonds) and are frequently used for calculations of properties (e.g. dipole moments, polarizabilities, etc.). For the Pople's basis sets the following notaton is used: n-ij+G, or n-ijk+G when 1 diffuse s-type and p-type gaussian with the same exponents are added to a standard basis set on heavy atoms. The n-ij++G, or n-ijk++G are obtained by adding 1 diffuse s-type and p-type gaussian on heavy atoms and 1 diffuse s-type gaussian on hydrogens. For example, the 6-31+G* represents (6311,311,1)/(31) or (11s,5p,1d/4s) -> [4s,3p,1d/2s]. The 6-311+G(2d1f,2p1d) stands for (63111,3111,11,1)/(311,11,1) split, or (12s,6p,2d,1f) -> [5s,2p,1d]. For more information about diffuse functions see, for example (Clark et al., 1983), (Del Bene, 1989), and (Frisch et al., 1984). To calculate total number of primitives/basis functions in your molecule, you sum up the number of primitives/basis functions for each partaking atom. As an example, let us compute the number of functions for H2SO3 molecule assuming the use of Gaussian90 program. The 6-311++G(3df,2p) basis set is used as an example. In this case the reduced set of d and f gaussians is used, i.e., 5 d-type functions and 7 f-type functions. It corresponds to the following contractions: S: (6311111,421111,111,1) (for sulphur, Gaussian90 defaults to McLean- Chandler basis set (631111,42111) for sulphur anion which is augmented with one diffuse s and one diffuse p function, and three d and one f polarization functions) O: (63111,3111,111,1) (this is 6-311G for oxygen augmented with one s- and one p-type diffuse function, and three d and one f polarization function) H: (3111,11) (this is 6-311G augmented with one diffuse s and two p-functions for polarization) Number of basis functions: - - - - - - - - - - - - - S: 7 s-type functions, 6*3 p-type functions, 3*5 d-type functions and 1*7 f-type functions O: 5 s-type functions, 4*3 p-type functions, 3*5 d-type functions and 1*7 f-type functions H: 4 functions of type s and 2*3 functions of type p (there are 3 p function for each p type contraction, i.e. p_x, p_y, p_z) H2SO3 = (4 + 2*3)*2 + (7 + 6*3 + 3*5 + 1*7) + (5 + 4*3 + 3*5 + 1*7)*3 = 184 Total number of gaussian primitives: - - - - - - - - - - - - - - - - - - S: 1*(6+3+1+1+1+1+1) + 3*(4+2+1+1+1+1) + 5*(1+1+1) + 7*1 = 66 O: 1*(6+3+1+1+1) + 3*(3+1+1+1) + 5*(1+1+1) + 7*1 = 52 H: 1*(3+1+1+1) + 3*(1+1) = 12 H2SO3 = 2*12 + 66 + 3*52 = 246 primitives. GENERAL CONTRACTIONS. TERMS AND NOTATION ======================================== Raffenetti (1973) introduced term "general contraction" for basis sets in which the same gaussian primitives can appear in several basis functions. In general contraction scheme, the basis functions are formed as different linear combinations of the same primitives. This is clearly in contrast with the segmented scheme described above. Please do not confuse general contrac- tions with a term "general basis set" used in some program manuals to denote "user defined segmented basis sets". General contractions have many advantages from the theoretical point of view. The most important is that they might be chosen to approximate true atomic orbitals which makes interpretation of coefficients in molecular orbitals meaningful. Also for correlated calculations their performance is praised (Almlof and Taylor, 1987; Almlof et al., 1988; Dunning, 1989) Secondly, they can be chosen in a more standard way than segmented contractions, either as true atomic orbitals obtained from Hartree- Fock calculations for the atom with uncontracted primitives as basis functions, or as Atomic Natural Orbitals (ANO). For description of ANO's consult papers by Almlof and coworkers or read appropriate chapter in Szabo and Ostlund, (1989). The only problem with general contractions is that only a few programs support them. The code for integral package is much more complicated in this case, since it has to work on a block of integrals at each time, to compute the contribution from the given primitive set only once. Of course, you can always enter general contractions as "user defined segmented basis sets," by repeating the same primitives over and over again in different contractions. This will cost you, however, immensely in computer time at the integral computation stage. Remember, the time required for calculating integrals is proportional to the 4th power in the number of gaussian primitives, and most programs assume that primitives entering different contractions are different. As an example, the general contractions of (8s4p) set of primitives for oxygen by Huzinaga et al., 1971 (taken from: Raffenetti, 1973). Exponents |------------ coefficients --------------------------| s-exponents 1s 2s s' s" 5.18664(+3) 1.95900(-3) 4.49000(-4) 0.00000 0.00000 7.77805(+2) 1.50290(-2) 3.38100(-3) 0.00000 0.00000 1.76161(+2) 7.38340(-2) 1.76630(-2) 0.00000 0.00000 4.93608(+1) 2.47316(-1) 6.05540(-2) 0.00000 0.00000 1.58205(+1) 4.73314(-1) 1.59948(-1) 0.00000 0.00000 5.51493 3.27039(-1) 1.46197(-1) 0.00000 0.00000 1.03159 1.93420(-2) -5.46581(-1) 0.00000 1.00000 3.06844(-1) -3.57900(-3) -5.84553(-1) 1.00000 0.00000 p-exponents 2p p' p" 1.78462(+1) 4.25100(-2) 0.00000 0.00000 3.88748 2.26972(-1) 0.00000 0.00000 1.05481 5.07788(-1) 0.00000 1.00000 2.77222(-1) 4.63550(-1) 1.00000 0.00000 In the table above, integer numbers in parantheses denote powers of 10 multiplying number in front of them. The set above can be described as (8s,4p) -> [4s,3p] contraction. Clearly, the notation giving the number of primitives in each contraction as (abcd...) is not really useful here. It is especially true with newer sets implementing general contractions, where each primitive has all nonzero coefficients in practically every column. EFFECTIVE CORE POTENTIALS (EFFECTIVE POTENTIALS) ================================================ It was known for a long time that core (inner) orbitals are in most cases not affected significantly by changes in chemical bonding. This prompted the development of Effective Core Potential (ECP) or Effective Potentials (EP) approaches, which allow treatment of inner shell electrons as if they were some averaged potential rather than actual particles. ECP's are not orbitals but modifications to a hamiltonian, and as such are very efficient computationally. Also, it is very easy to incorporate relativistic effects into ECP, while all-electron relativistic computations are very expensive. The relativistic effects are very important in describing heavier atoms, and luckily ECP's simplify calculations and at the same time make them more accurate with popular non-relativistic ab initio packages (provided that such packages have support for ECP's). The core potentials can only be specified for shells that are filled. For the rest of electrons (i.e. valence electrons), you have to provide basis functions. These are special basis sets optimized for the use with specific ECP's. These basis sets are usually listed in original papers together with corresponding ECP's. Some examples of papers describing ECP's: (Durand and Bartelat, 1975), (Hay and Wadt, 1985ab), (Hurley et al., 1986), (Pacios and Christensen, 1985), (Stevens ey al., 1984), (Wadt and Hay, 1985), (Walace et al., 1991). The ECP are tabulated in the literature as parameters of the following expansion: ECP(r) = sum (i=1 to M) { d_i*r^(n_i)*exp[-zeta_i*r^2] } where M is the number of terms in the expansion, d_i is a coefficient for each term, r denotes distance from nucleus, n_i is a power of r for the i-th term, and zeta_i represents the exponent for the i-th term. To specify ECP for a given atomic center, you need to include typically: the number of core electrons that are substituted by ECP, the largest angular momentum quantum number included in the potential (e.g., 1 for s only, 2 for s and p, 3 for s, p, and d; etc.), and number of terms in the "polynomial gaussian expansion" shown above. For each term in this expansion you need to specify: coefficient (d_i), power of r (n_i) and exponent in the gaussian function (zeta_i). Also you need to enter basis set for valence electrons specific to this potential. As a result of applying the ECP's you drastically reduce number of needed basis functions, since only functions for valence electrons are required. In many cases, it would simply be impossible to perform some calculations on systems involving heavier elements without ECP's (try to calculate number of functions in TZ2P basis set for e.g. U, and you will know why). ---