- Email: [email protected]
A universal basis set to be used along with pseudopotentials
21November1997
CHEMICAL PHYSICS LETTERS ELSEVIER
Chemical Physics Letters 279 (1997) 396-402
A universal basis set to be used along with pseudopotentials Marcelo Giordan 1, Rog6rio Custodio *, Nelson H. Morgon Universidade Estadual de Campinas, Instituto de Qulmica, Departamento de Ffsico-Qulmica, Caixa Postal 6154, 13083-970 Campinas, S~o Paulo, Brazil
Received 14 April 1997; in final form 4 September 1997
Abstract Universal basis sets from all-electron calculations can be adapted to be used along with pseudopotentials by the generator coordinate method (GCM). Calculations carried out with Hay and Wadt or Steven, Bash and Krauss pseudopotentials along with a universal basis set are in agreement with all-electron calculations and also with improved basis sets specially developed to be used with pseudopotential. The support provided by GCM allows a simple procedure to enlarge or reduce the size of the universal basis set as well as a systematic procedure to include polarization functions. © 1997 Elsevier Science B.V.
1. Introduction Universal basis sets have been developed for allelectron calculations in the last few years [1-12]. Gaussian and Slater functions provided different universal basis sets comprising either a limited or a large number of atoms including almost all the elements of the periodic table. A theoretical support for the universal basis sets can be found in a method called the generator coordinate method ( G C M ) [ 7 12]. This method suggests that one electron functions can be represented as an integral transform:
~ti(l) = f(} f/( O~) I~bi( OL,1 ) d of ,
(1)
where fi and (~i are the weight function and genera-
" Corresponding author. Fax: + 55-19-239-3805. i Permanent address: Departamento de Metodologia do Ensino e Educa~j~,oComparada, Faculdade de Educa~jao,Universidade de S~_o Paulo, Av. da Universidade, 308, 05508-900 - S~.o Paulo, S~o Paulo, Brazil.
tor function, respectively, and ce is the generator coordinate. For individual atoms the one electron functions are integrated numerically and the space of exponents is truncated for each particular symmetry of ~b. A particular mesh is chosen after analysis of the accuracy of a single property or by a set of different ones. The behavior of f~ has been often used to establish the cutoff of the exponents in the valence and inner electronic region. Tests of weight functions obtained with Gaussian functions showed a continuous and smooth tendency converging to zero at the boundaries of the c~ space. These properties have been used either to model or to improve basis sets [13-18]. An important aspect in this approach is that the ce exponents are seen as discretization points required to produce an accurate numerical integration of Eq. (1). In this sense a universal basis set is just an adequate choice of a large mesh suitable to integrate simultaneously one electron functions of different atoms. Several procedures can be used to define the discretization points. However, the eventempered basis sets [19-21], a particular discretiza-
0009-2614/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved. PI1 S0009-2614(97)01058-0
M. Giordan et al. / Chemical Physics Letters 279 (1997) 396-402
tion technique of GCM [14], have been used as the simplest choice to model universal basis sets [5,11]. This procedure was applied successfully to model different universal basis sets for all-electron atoms [7-12]. The limited use of the available universal basis sets in molecular calculations is consequence of their large size. For example, Malli et al. [12] developed an accurate universal Gaussian basis set to be used in atomic calculations including relativistic effects. The basis set for He (Z = 2) to No (Z = 102) required (32s29p20d15f) to reach the numerical accuracy in the energy to a part in l09 for lighter atoms and to a part in 108 for heavier elements. It is clear that for molecular applications such large basis set should be contracted or adapted for the atoms in the molecular environment in an attempt to reduce its size. The need for a large number of primitives is much more closely related to the careful energetic balance in the core region than in the valence one. A simple way to avoid a large number of primitives in the inner region is to change the core atomic orbitals by a pseudopotential. Recent comparisons of the weight functions calculated with all-electrons and pseudopotentials for atoms showed some interesting results [16-18]. While in all-electron calculations the cutoff of the inner most region is strongly dependent of the property used to model the basis set, the use of a pseudopotential like Hay and Wadt (HW) [18-24] provided a sharp inner exponent above which the basis set does not produce any significant effect on the calculation. The second point was a complete similarity between the weight functions calculated with all-electrons and pseudopotential in the valence region when no relativistic effect was included [16,18]. In calculations where the relativistic effects were taken into account the weight functions calculated from both methods presented some differences in the valence region [ 16,18]. The objective of this Letter is to generalize the results obtained in previous papers adapting a universal Gaussian basis set to be used along with pseudopotentials. Malli's universal basis set is adapted along with Hay and Wadt (HW) [23,24], and Steven, Basch and Krauss (SBK) [25] pseudopotentials for atoms from He to No. The accuracy of the basis sets is tested for some selected atoms (excitation of energies of V, Cr, Mn, Fe, Co and Ni) and molecules (diatomic halides and PbCI2). A simple
397
procedure based on GCM is presented to reduce the basis set size for specific atoms and different molecular environments. The procedure outlined below can in principle be used for other all-electron universal basis set as well as for different pseudopotentials.
2. Computational methods Universal basis set for all-electron calculations can be developed using two basic information from GCM: (a) the tendency of Roothaan's linear combination coefficients of the basis set expansion (LCAO), and (b) the choice of properties monitoring the basis set accuracy. The LCAO coefficients are an approximation of the true weight functions. A convenient way to visualize the approximated weight functions is to relabel the o~ exponents as ~1i = ln(ai) and to plot the LCAO coefficients versus the respective ~Qi- Fig. l a shows the weight functions for the valence s function for F to I calculated at the Hartree-Fock level using Malli's universal Gaussian geometric basis set (32s29p20d15f). This universal geometric basis set was obtained using ~Q,, = - 3.84, A~Q = 0.72 for all the symmetry species and the respective number of primitives indicated above. The solid lines in Fig. 1 are the s valence weight functions from all-electron calculation and the dashed lines represent the equivalent s weight functions calculated with the same basis set, but replacing the inner electrons by the SBK pseudopotential [25]. it can be observed that the LCAO coefficients lk~r the four halogens present a smooth behavior. The approximate weight functions are well described by the universal basis set, vanishing at the boundaries of the c~ space in both calculations. Some of the inner most exponents were removed in Fig. 1. The fluorine weight functions are well described in the valence region by exponents greater than ,Q,, = - 2 . 4 0 (o~,, = 0.090718) while iodine is described by exponents starting from .(2,, = - 3 . 1 2 (c~, = 0.044157). Therefore a single basis set to be used in the four halogens should start from the iodine most diffuse exponent, in this case g~, = - 3 . 1 2 ( a o = 0.044157). The inner most function can be determined in the same way. In other words, the largest exponent responsible ['or significant changes in a reference property of any of the four atoms would be the inner most limit of the
M. Giordan et al. / Chemical Physics Letters 279 (1997) 396-402
398
region enlarges notably the final basis set. However, the number of primitives can be reduced drastically changing the inner electronic distribution to a pseudopotential. Fig. 1 shows a significant difference between weight functions calculated with the allelectron and pseudopotential for the valence s atomic orbitals of halogens. The curves related with the pseudopotential remains nearly unchanged in going from a given value of ,(2 (or a ) towards larger values. A comparison between all-electron and pseudopotential curves indicates clearly where the pseudopotential starts to take effect and thus presents an inner point at which to cut the basis set. The appropriate inner most limit of the s universal mesh can be
universal basis set. However, Fig. 1 shows that the cutoff of the inner region is difficult to determine for all-electron calculations. The LCAO coefficients associated with large exponents present a set of values close to zero, but significant for properties like the electronic energy. An adequate choice of the inner most limit for all-electron calculation depends on an analysis of very small coefficients associated with the convergence of one or more properties sensitive to the electronic distribution around the nuclear region• The electronic energy is frequently used, although other properties can also be employed simultaneously [11]. A large number of primitives needed to describe this
0,8
F
0,4 0,0
-0,4 '
0,8-
0,4~
I
I
,
I
'
I
~
i
I'
~
1
I
'
I
CI i
0,0 -~
~.. -0,4 0,8-
'
I
'
'
!
"
'
BF ,o' °', °
0,4 0,0
Wt
-0,4 '
0,8
I
I
'
'
.'o,
I
I
0,4 o.
• •
0,0
"
,
{
o."
-0,4 '
-5,0
I
I
-2,5
0,0
'
I
I
l
2,5
5,0
7.5
In(cO Fig. l. Weight functions for the valence s atomic orbital of halogens obtained from all-electron calculations (solid lines) and pseudopotential (dotted lines) with M a l l i ' s universal basis set. Only the first 17 points are shown in this figure.
M. Giordan et al. / Chemical Physics Letters 279 (1997) 396-402 Table 1 Basis set exponents (c~) of the universal Gaussian type functions for s, p, d and f symmetries 0.010462 0.021494 0.044157 0.090718 0.186374 0.382893 0.786628 1.616074 3.320117 6.820958 14.013204
defined as the largest ~0 obtained from the analysis of the weight functions calculated with pseudopotential. The cutoff in the inner region for the p, d and f meshes can be the same used for the s mesh. This method was used to adapt Malli's universal basis set in atoms using HW and SBK pseudopotenrials. The analysis of the weight functions from He to No provided a (1 lsl lpl ldl If) basis set with g2o = - 4 . 5 6 , Ag2 = 0.72 and N = 11 for the s, p, d and f symmetry functions. Table 1 shows the exponents for the adapted universal Gaussian basis set. The complete original universal Gaussian basis set was reduced from 338 to 176 primitives.
3. Results and discussion Calculations of the atomic excitation energies of some transition metals comprise an interesting test for the performance of the universal basis set. In
399
transition metal calculations one of the main problems correspond to the correct description of the d mesh, since it has to be included as an adequate basis set to describe the valence d atomic orbitals. It can be found in the literature different sets of d type basis functions. The calculated excitation energies are very susceptible to the set of d optimized exponents. Table 2 shows the excitation energies for six transition elements (V, Cr, Mn, Fe, Co and Ni) calculated with the universal basis set and HW pseudopotential (2nd column), and also calculated by other methods. The 3rd and 4th columns present results calculated using nonrelativistic pseudopotential and (8s7p6dlf)/[6s5p3dlf] basis set [26]. The 5th column corresponds to the all-electron calculations carried out by Hay [27] using optimized d functions for each atom in an attempt to improve the calculated excitation energies with respect to the numerical results. The 6th column shows the results calculated by Goddard et al. [28] also using all-electron Hartree-Fock calculations and optimized d functions for each atom. The errors in the excitation energies calculated with UB and HW pseudopotential are systematically 0.2-0.3 eV lower than the numerical results. The data presented in the 3rd column using a different pseudopotential show the opposite trend, while a better pseudopotential (4th column) and Hay's all-electron results fluctuate around the numerical results. Goddard's results are also larger than the reference data. These features emphasize the great dependence of the calculated properties upon the pseudopotential or the specific way the d meshes were optimized. It is worth noting the excellent agreement provided by the calculations
Table 2 Calculated excitation energies (in eV) from H a r t r e e - F o c k calculations
V (3d34s 2 ~ 3d 5) Cr (3d54s I -~ 3d 6) Mn (3dS4s 2 ~ 3d 7 ) Fe (3d64s: -~ 3d ~) Co (3d74s 2 -~ 3d ~) Ni (3d~4s 2 -+ 3d m)
This work ~
Sefit b
Merit b
Hay ~
Goddard d
Numerical HF ~
3.09 5.44 8.93 7.27 6.86 5.30
3.50 6.03 9.49 7.83 7.47 5.96
3.29 5.80 9.14 7.48 7.06 5.49
3.31 5.79 9.18 7.47 7.06 5.47
3.32 5.81 9.21 7.51 7.07 5.56
3.27 5.75 9.15 7.46 7.05 5.47
~' The universal basis set and Hay and Wadt pseudopotential were used in all calculations. u Sefit (single-electron-fit nonrelativistic pseudopotential) and mefit (multielectron-fit nonrelativistic pseudopotential) from Ref. [26]. Ref. [27]. a Ref. [28].
400
M. Giordan et al. / Chemical Physics Letters 279 (1997) 396-402
with the universal basis set and H W pseudopotential with respect to the all-electron numerical results. Although the universal basis set (1 l s l l p l l d l l f ) can be used in any atom, it is not computationally efficient to employ the complete basis functions in molecular calculations. In this sense the analysis of the weight functions can be used to minimize the universal basis set size in the molecular environment as shown above or as described in detail in ref.[18] where theoretical aspects on nonuniversal basis sets to be used along with pseudopotentials are also discussed. Another important procedure is related to the inclusion of polarization functions. Some of the symmetry functions included in the universal basis functions can be considered in specific cases as polarization functions and can be reduced drastically. The diatomic hydrides and PbCI 2 molecules are presented to illustrate the possible technique to minimize the basis set size. For the diatomic halogens represented by the SBK pseudopotentiai, the universal basis set can be restricted to the (1 lsl lp) basis functions, the symmetry functions associated to the valence atomic orbitals. Some of the functions with higher angular momentum selected from the universal basis set can be included as polarization functions. The choice of the appropriate exponents of the polarization functions can be determined considering the improvement of one or more molecular properties. In this example, d type functions are included considering the minimization of the electronic energy at the equilibrium geometry calculated at the Hartree-Fock level of theory. In the sequence, the analysis of the magnitude of the L C A O coefficients (weight functions) is carried out for the occupied molecular orbitals. The number of d type functions required to provide the minimum energy criterion changes from halogen to halogen. Iodine and bromine requires a set of four d type functions with ~o = - 1.68. Chlorine also requires four d type functions starting from 12o = - 0 . 9 6 , while fluorine uses three d type functions with 120 = - 0 . 9 6 . All the calculations use sets of 5d type functions (dxy, d ~ , dy:, d:z and d~2_,2). The results presented in Table 3, calculated with the universal basis set for the four dihalides, are in excellent agreement with available data from the literature calculated by other all-electron methods or with SBK pseudopotential and adapted basis func-
Table 3 Electronic properties o f the X 2 molecules ( G = equilibrium geometry, to~ = h a r m o n i c frequency and A E x = E x z - 2 E x ) in the ground state calculated with the pseudopotential and results from the literature Molecule
Method
r~/A
F2
HF a MP2 a MP4 ~ DFT h exp. ~
1.34 1.34 1.43 1.41 1.44
986.2 880.8 1040.0 917.0
- 1.67 1.55 1.54 2.62 1.72
HF ~ MP2 a M P 4 ~' DFT b AE J ECP ~ exp. c
1.99 1.99 2.02 2.10 2.06 2.05 1.99
591.3 535.4 494.0 541.1 528.2 565.0
0.38 1.92 2.19 2.21 2.30 2.18 2.48
HF a MP2" MP4 a AE d ECP e exp. ~
2.30 2.30 2.33 2.33 2.34 2.28
323.9 306.5 321.7 312.9 325.3
0.28 1.59 1.87 2.09 2.02 1.97
HF a MP2 a MP4 ~ DFT b exp. ~
2.68 2.69 2.71 2.65 2.67
220.0 209.0 221.0 215.0
0.49 1.68 1.69 2.17 1.54
CI 2
Br 2
12
o~e / c m -
I
A E x/eV
a This work. hCalculation with D F T using the pseudopotential and basis set of D Z V P quality plus nonlocal corrections o f Becke and Perdew
[30]. Ref. [31]. M C S C F calculation e m p l o y i n g T Z basis sets with all electrons
[32]. MCSCF calculation employingTZ basis sets with the pseudopotential of Hay and Wadt [32]. tions. The deviation can be associated either with the pseudopotential or with the original universal basis set. Larger universal basis set can be developed, yielding results in the limit of saturation of the basis sets [15]. The parameters for the basis set can be defined from the largest and smallest exponents of the universal basis set. A12 is then defined as: A12(k) = [12~(k) - 12o(k)]/(Nk - 1), where 12f(k) and 12o(k) are the logarithm of the largest and smallest exponents with symmetry k, and N k is the number of primitives in the mesh. N k can be large so as to saturate the universal basis set.
401
M. Giordan et al. / Chemical Physics Letters 279 (1997) 396-402
Table 4 Molecularproperties of the PbC12 molecule(r~ = equilibriumgeometry, 0 = equilibriumbond angle,/~ = dipole momentsand co= harmonic frequencies) in the ground state calculatedwith the universalbasis set and HW pseudopotentialand experimentalresults from the literature r¢/A
0/deg
/x/debye
~oI/cm- i
co~/cm t
o)~/cm- i
UB-MP2 ' UB-SCF ~' HW-SCF •
2,43 2.43 2,51
99.7 99,5 100.9
4.834 5.178 6.657
105.0 114.7 100.0
299.9 296.1 293.6
319.1 319.1 305.7
exp.
2.46
101:4
307 + 3 ~l 369 ~
329 +_3 '~ 365 ~
80 + I0 ~1 94 ~
MP2 calculationusing the universalbasis set with Hay and Wadt pseudopotential. b Hartree-Fock calculationusing the universalbasis set with Hay and Wadt pseudopotential. " Hartree-Fock calculationusing the Hay and Wadt pseudopotentialand the original basis set. d Ref. [33]. ~"Ref. [34].
A final application is carried out for PbCI 2, where relativistic effects are important to appropriately describe some molecular properties. The HW pseudopotential is used in these calculations, since the SBK pseudopotential does not take into account relativistic effects for heavier atoms. The s and p basis set for C1 is the same employed in the calculation of CI 2, discussed above. Three polarization functions were included with /2 o = - 0 . 9 6 . The basis set for Pb consists of the (1 l s l lp) universal mesh and two d polarization functions with 12o = - 3 . 8 4 , determined using the same criteria discussed for the diatomic halides. To minimize computational costs the adapted universal basis set for Pb and C1 were contracted by the D u n n i n g ' s segmented method [29]. The basis set for Pb was contracted from (1 lsl lp2d) to [5s5p2d] and the basis function for CI was contracted from ( l l s l l p 3 d ) to [9s9p3d]. The lost of energy after the basis set contraction was lower than 1 mE H with respect to the same calculation using the uncontracted basis set. Table 4 shows the results calculated with the universal basis set along with HW pseudopotential, with the HW pseudopotential and its original basis set and the experimental results from the literature. The calculated properties are in good agreement with experimental data. An interesting aspect can be observed when comparison is made with the results calculated via the original HW basis set and the universal basis set along with the HW pseudopotential. The C1-Pb-C1 bond angle calculated with the original HW basis set is closer to the experimental result than the universal basis set
data using the same pseudopotential. However, significant differences can be observed when other calculated molecular properties are compared. For instance, the dipole moment calculated using both basis set at the Hartree-Fock level of theory differs in almost 1.5 debye, a remarkable difference in the charge distribution. The absence of polarization functions and the small size of the original HW basis set can be pointed out as the responsible for the significant differences.
4. Conclusion In conclusion, the generator coordinate method proved to be adequate to adapt all-electron universal basis set to atoms represented by pseudopotentials. The use of the resulting universal basis set with atoms described by pseudopotentials yields results compatible with all-electron calculations. This paper indicates the possibility to expand or to reduce the size of the universal basis set either in the valence or in the inner region, using as criterion the analysis of the behavior of weight functions of the atomic or molecular systems. The polarization functions can also be easily obtained from the UB mesh with primitives of higher angular momentum through the analysis of the magnitude of the LCAO coefficients and some other molecular property of interest like the total electronic energy.
402
M. Giordan et a l . / Chemical Physics Letters 279 (1997) 396-402
Acknowledgements We acknowledge financial support from FAPESP (Funda~o de Ampfiro h Pesquisa do Estado de S~.o Paulo), CNPq (Conselho Nacional de Desenvolvimento Cientffico e Tecnol6gico) and CAPES (Coordenadoria de Aperfei~joamento de Pessoal de Ensino Superior). We also thank the computational facilities of Cenapad (Centro Nacional de Processamento de Alto Desempenho) - SP.
References [1] D.M. Silver, S. Wilson, W.C. Newport, Int. J. Quantum Chem. 14 (1978) 635. [2] D.M. Silver, W.C. Newport, Chem. Phys. Lett. 57 (1978) 421. [3] D.M. Silver, S. Wilson, J. Chem. Phys. 69 (1978) 3787. [4] S. Wilson, Adv. Chem. Phys. 67 (1987) 440. [5] D. Moncrieff, S. Wilson, J. Phys. B: At. Mol. Opt. Phys. 27 (1994) 1. [6] P.G. Mezey, Theor. Chim. Acta 53 (1979) 183. [7] J.R. Mohallem, Z. Phys. D: At. Mol. Clusters 3 (1986) 339. [8] J.R. Mohallem, M. Trsic, J, Chem. Phys. 86 (1987) 5043. [9] H.F.M. da Costa, M. Trsic, J.R. Mohallem, Mol. Phys. 62 (1987) 91. [10] A.B.F. da Silva, H.F.M. da Costa, M. Trsic, Mol. Phys. 68 (1989) 433. [11] R. Custodio, J.D. Goddard, M. Giordan, N.H. Morgon, Can. J. Chem. 70 (1992) 580. [12] G.L. Malli, A.B.F. da Silva, Y. lshikawa, Phys. Rev. 47 (1993) 143.
[13] R. Custodio, J.D. Goddard, J. Mol. Struct. (Theochem) 277 (1992) 263. [14] R. Custodio, M. Giordan. N.H. Morgon, J.D. Goddard, lnt. J. Quantum Chem. 42 (1992) 411. [15] R. Custodio, J.D. Goddard, J. Mol. Struct. (Theochem) 281 (1993) 75. [16] R. Custodio, W.M. Davis, J.D. Goddard, J, Mol. Struct. (Theochem) 315 (1994) 163. [17] N.H. Morgon, R. Custodio, J.M. Rivetos, Chem. Phys. Lett. 235 (1995) 436. [18] M. Giordan, R. Custodio, J. Comp. Chem. (in press). [19] R.D. Barrio. K. Ruedenberg, J. Chem. Phys. 59 (1973) 5936. [20] D.F, Feller, K. Ruedenberg, Theor. Chim. Acta 52 (1979) 231, [21] M.W. Schmidt, K. Ruedenberg, J. Chem. Phys. 71 (1979) 3951. [22] P.J. Hay, W.R. Wadt, J. Chem. Phys. 82 (1985) 270. [23] W.R. Wadt, P.J. Hay, J. Chem. Phys. 82 (1985) 284. [24] P.J. Hay, W.R. Wadt, J. Chem. Phys. 82 (1985) 299. [25] W. Stevens, H. Basch, J. Krauss. J. Chem, Phys. 81 (1984) 6026. [26] M. Dolg, U. Wedig, H. Stoll, H. Preuss, J, Chem. Phys. 66 (1987) 4377. [27] P.J. Hay, J. Chem. Phys. 66 (1977) 4377. [28] A.K. Rappe, T.A. Smedley, W.A. Goddard IlI, J. Phys. Chem. 85 (1981) 2607. [29] T.H. Dunning, J. Chem. Phys. 53 (1970) 2823. [30] H. Chert. M. Krasowski, G.J. Fitzgerald, J. Chem. Phys. 98 (1993) 8710. [31] K. Huber, G. Herzberg, Molecular Spectra and Molecular Structure IV, Constants of Diatomic Molecules, Van Nostrand Reinhold, New York, 1979. [32] M. Klobukovski, Theor. Chim. Acta 83 (1992) 239, [33] L. Brewer, G.R. Somayajulu, E. Brackett, Chem. Rev. 63 (1963) 111. [34] M. Dupuis, J. Rys, H.F. King, J. Chem. Phys. 65 (1976) 111.
CHEMICAL PHYSICS LETTERS ELSEVIER
Chemical Physics Letters 279 (1997) 396-402
A universal basis set to be used along with pseudopotentials Marcelo Giordan 1, Rog6rio Custodio *, Nelson H. Morgon Universidade Estadual de Campinas, Instituto de Qulmica, Departamento de Ffsico-Qulmica, Caixa Postal 6154, 13083-970 Campinas, S~o Paulo, Brazil
Received 14 April 1997; in final form 4 September 1997
Abstract Universal basis sets from all-electron calculations can be adapted to be used along with pseudopotentials by the generator coordinate method (GCM). Calculations carried out with Hay and Wadt or Steven, Bash and Krauss pseudopotentials along with a universal basis set are in agreement with all-electron calculations and also with improved basis sets specially developed to be used with pseudopotential. The support provided by GCM allows a simple procedure to enlarge or reduce the size of the universal basis set as well as a systematic procedure to include polarization functions. © 1997 Elsevier Science B.V.
1. Introduction Universal basis sets have been developed for allelectron calculations in the last few years [1-12]. Gaussian and Slater functions provided different universal basis sets comprising either a limited or a large number of atoms including almost all the elements of the periodic table. A theoretical support for the universal basis sets can be found in a method called the generator coordinate method ( G C M ) [ 7 12]. This method suggests that one electron functions can be represented as an integral transform:
~ti(l) = f(} f/( O~) I~bi( OL,1 ) d of ,
(1)
where fi and (~i are the weight function and genera-
" Corresponding author. Fax: + 55-19-239-3805. i Permanent address: Departamento de Metodologia do Ensino e Educa~j~,oComparada, Faculdade de Educa~jao,Universidade de S~_o Paulo, Av. da Universidade, 308, 05508-900 - S~.o Paulo, S~o Paulo, Brazil.
tor function, respectively, and ce is the generator coordinate. For individual atoms the one electron functions are integrated numerically and the space of exponents is truncated for each particular symmetry of ~b. A particular mesh is chosen after analysis of the accuracy of a single property or by a set of different ones. The behavior of f~ has been often used to establish the cutoff of the exponents in the valence and inner electronic region. Tests of weight functions obtained with Gaussian functions showed a continuous and smooth tendency converging to zero at the boundaries of the c~ space. These properties have been used either to model or to improve basis sets [13-18]. An important aspect in this approach is that the ce exponents are seen as discretization points required to produce an accurate numerical integration of Eq. (1). In this sense a universal basis set is just an adequate choice of a large mesh suitable to integrate simultaneously one electron functions of different atoms. Several procedures can be used to define the discretization points. However, the eventempered basis sets [19-21], a particular discretiza-
0009-2614/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved. PI1 S0009-2614(97)01058-0
M. Giordan et al. / Chemical Physics Letters 279 (1997) 396-402
tion technique of GCM [14], have been used as the simplest choice to model universal basis sets [5,11]. This procedure was applied successfully to model different universal basis sets for all-electron atoms [7-12]. The limited use of the available universal basis sets in molecular calculations is consequence of their large size. For example, Malli et al. [12] developed an accurate universal Gaussian basis set to be used in atomic calculations including relativistic effects. The basis set for He (Z = 2) to No (Z = 102) required (32s29p20d15f) to reach the numerical accuracy in the energy to a part in l09 for lighter atoms and to a part in 108 for heavier elements. It is clear that for molecular applications such large basis set should be contracted or adapted for the atoms in the molecular environment in an attempt to reduce its size. The need for a large number of primitives is much more closely related to the careful energetic balance in the core region than in the valence one. A simple way to avoid a large number of primitives in the inner region is to change the core atomic orbitals by a pseudopotential. Recent comparisons of the weight functions calculated with all-electrons and pseudopotentials for atoms showed some interesting results [16-18]. While in all-electron calculations the cutoff of the inner most region is strongly dependent of the property used to model the basis set, the use of a pseudopotential like Hay and Wadt (HW) [18-24] provided a sharp inner exponent above which the basis set does not produce any significant effect on the calculation. The second point was a complete similarity between the weight functions calculated with all-electrons and pseudopotential in the valence region when no relativistic effect was included [16,18]. In calculations where the relativistic effects were taken into account the weight functions calculated from both methods presented some differences in the valence region [ 16,18]. The objective of this Letter is to generalize the results obtained in previous papers adapting a universal Gaussian basis set to be used along with pseudopotentials. Malli's universal basis set is adapted along with Hay and Wadt (HW) [23,24], and Steven, Basch and Krauss (SBK) [25] pseudopotentials for atoms from He to No. The accuracy of the basis sets is tested for some selected atoms (excitation of energies of V, Cr, Mn, Fe, Co and Ni) and molecules (diatomic halides and PbCI2). A simple
397
procedure based on GCM is presented to reduce the basis set size for specific atoms and different molecular environments. The procedure outlined below can in principle be used for other all-electron universal basis set as well as for different pseudopotentials.
2. Computational methods Universal basis set for all-electron calculations can be developed using two basic information from GCM: (a) the tendency of Roothaan's linear combination coefficients of the basis set expansion (LCAO), and (b) the choice of properties monitoring the basis set accuracy. The LCAO coefficients are an approximation of the true weight functions. A convenient way to visualize the approximated weight functions is to relabel the o~ exponents as ~1i = ln(ai) and to plot the LCAO coefficients versus the respective ~Qi- Fig. l a shows the weight functions for the valence s function for F to I calculated at the Hartree-Fock level using Malli's universal Gaussian geometric basis set (32s29p20d15f). This universal geometric basis set was obtained using ~Q,, = - 3.84, A~Q = 0.72 for all the symmetry species and the respective number of primitives indicated above. The solid lines in Fig. 1 are the s valence weight functions from all-electron calculation and the dashed lines represent the equivalent s weight functions calculated with the same basis set, but replacing the inner electrons by the SBK pseudopotential [25]. it can be observed that the LCAO coefficients lk~r the four halogens present a smooth behavior. The approximate weight functions are well described by the universal basis set, vanishing at the boundaries of the c~ space in both calculations. Some of the inner most exponents were removed in Fig. 1. The fluorine weight functions are well described in the valence region by exponents greater than ,Q,, = - 2 . 4 0 (o~,, = 0.090718) while iodine is described by exponents starting from .(2,, = - 3 . 1 2 (c~, = 0.044157). Therefore a single basis set to be used in the four halogens should start from the iodine most diffuse exponent, in this case g~, = - 3 . 1 2 ( a o = 0.044157). The inner most function can be determined in the same way. In other words, the largest exponent responsible ['or significant changes in a reference property of any of the four atoms would be the inner most limit of the
M. Giordan et al. / Chemical Physics Letters 279 (1997) 396-402
398
region enlarges notably the final basis set. However, the number of primitives can be reduced drastically changing the inner electronic distribution to a pseudopotential. Fig. 1 shows a significant difference between weight functions calculated with the allelectron and pseudopotential for the valence s atomic orbitals of halogens. The curves related with the pseudopotential remains nearly unchanged in going from a given value of ,(2 (or a ) towards larger values. A comparison between all-electron and pseudopotential curves indicates clearly where the pseudopotential starts to take effect and thus presents an inner point at which to cut the basis set. The appropriate inner most limit of the s universal mesh can be
universal basis set. However, Fig. 1 shows that the cutoff of the inner region is difficult to determine for all-electron calculations. The LCAO coefficients associated with large exponents present a set of values close to zero, but significant for properties like the electronic energy. An adequate choice of the inner most limit for all-electron calculation depends on an analysis of very small coefficients associated with the convergence of one or more properties sensitive to the electronic distribution around the nuclear region• The electronic energy is frequently used, although other properties can also be employed simultaneously [11]. A large number of primitives needed to describe this
0,8
F
0,4 0,0
-0,4 '
0,8-
0,4~
I
I
,
I
'
I
~
i
I'
~
1
I
'
I
CI i
0,0 -~
~.. -0,4 0,8-
'
I
'
'
!
"
'
BF ,o' °', °
0,4 0,0
Wt
-0,4 '
0,8
I
I
'
'
.'o,
I
I
0,4 o.
• •
0,0
"
,
{
o."
-0,4 '
-5,0
I
I
-2,5
0,0
'
I
I
l
2,5
5,0
7.5
In(cO Fig. l. Weight functions for the valence s atomic orbital of halogens obtained from all-electron calculations (solid lines) and pseudopotential (dotted lines) with M a l l i ' s universal basis set. Only the first 17 points are shown in this figure.
M. Giordan et al. / Chemical Physics Letters 279 (1997) 396-402 Table 1 Basis set exponents (c~) of the universal Gaussian type functions for s, p, d and f symmetries 0.010462 0.021494 0.044157 0.090718 0.186374 0.382893 0.786628 1.616074 3.320117 6.820958 14.013204
defined as the largest ~0 obtained from the analysis of the weight functions calculated with pseudopotential. The cutoff in the inner region for the p, d and f meshes can be the same used for the s mesh. This method was used to adapt Malli's universal basis set in atoms using HW and SBK pseudopotenrials. The analysis of the weight functions from He to No provided a (1 lsl lpl ldl If) basis set with g2o = - 4 . 5 6 , Ag2 = 0.72 and N = 11 for the s, p, d and f symmetry functions. Table 1 shows the exponents for the adapted universal Gaussian basis set. The complete original universal Gaussian basis set was reduced from 338 to 176 primitives.
3. Results and discussion Calculations of the atomic excitation energies of some transition metals comprise an interesting test for the performance of the universal basis set. In
399
transition metal calculations one of the main problems correspond to the correct description of the d mesh, since it has to be included as an adequate basis set to describe the valence d atomic orbitals. It can be found in the literature different sets of d type basis functions. The calculated excitation energies are very susceptible to the set of d optimized exponents. Table 2 shows the excitation energies for six transition elements (V, Cr, Mn, Fe, Co and Ni) calculated with the universal basis set and HW pseudopotential (2nd column), and also calculated by other methods. The 3rd and 4th columns present results calculated using nonrelativistic pseudopotential and (8s7p6dlf)/[6s5p3dlf] basis set [26]. The 5th column corresponds to the all-electron calculations carried out by Hay [27] using optimized d functions for each atom in an attempt to improve the calculated excitation energies with respect to the numerical results. The 6th column shows the results calculated by Goddard et al. [28] also using all-electron Hartree-Fock calculations and optimized d functions for each atom. The errors in the excitation energies calculated with UB and HW pseudopotential are systematically 0.2-0.3 eV lower than the numerical results. The data presented in the 3rd column using a different pseudopotential show the opposite trend, while a better pseudopotential (4th column) and Hay's all-electron results fluctuate around the numerical results. Goddard's results are also larger than the reference data. These features emphasize the great dependence of the calculated properties upon the pseudopotential or the specific way the d meshes were optimized. It is worth noting the excellent agreement provided by the calculations
Table 2 Calculated excitation energies (in eV) from H a r t r e e - F o c k calculations
V (3d34s 2 ~ 3d 5) Cr (3d54s I -~ 3d 6) Mn (3dS4s 2 ~ 3d 7 ) Fe (3d64s: -~ 3d ~) Co (3d74s 2 -~ 3d ~) Ni (3d~4s 2 -+ 3d m)
This work ~
Sefit b
Merit b
Hay ~
Goddard d
Numerical HF ~
3.09 5.44 8.93 7.27 6.86 5.30
3.50 6.03 9.49 7.83 7.47 5.96
3.29 5.80 9.14 7.48 7.06 5.49
3.31 5.79 9.18 7.47 7.06 5.47
3.32 5.81 9.21 7.51 7.07 5.56
3.27 5.75 9.15 7.46 7.05 5.47
~' The universal basis set and Hay and Wadt pseudopotential were used in all calculations. u Sefit (single-electron-fit nonrelativistic pseudopotential) and mefit (multielectron-fit nonrelativistic pseudopotential) from Ref. [26]. Ref. [27]. a Ref. [28].
400
M. Giordan et al. / Chemical Physics Letters 279 (1997) 396-402
with the universal basis set and H W pseudopotential with respect to the all-electron numerical results. Although the universal basis set (1 l s l l p l l d l l f ) can be used in any atom, it is not computationally efficient to employ the complete basis functions in molecular calculations. In this sense the analysis of the weight functions can be used to minimize the universal basis set size in the molecular environment as shown above or as described in detail in ref.[18] where theoretical aspects on nonuniversal basis sets to be used along with pseudopotentials are also discussed. Another important procedure is related to the inclusion of polarization functions. Some of the symmetry functions included in the universal basis functions can be considered in specific cases as polarization functions and can be reduced drastically. The diatomic hydrides and PbCI 2 molecules are presented to illustrate the possible technique to minimize the basis set size. For the diatomic halogens represented by the SBK pseudopotentiai, the universal basis set can be restricted to the (1 lsl lp) basis functions, the symmetry functions associated to the valence atomic orbitals. Some of the functions with higher angular momentum selected from the universal basis set can be included as polarization functions. The choice of the appropriate exponents of the polarization functions can be determined considering the improvement of one or more molecular properties. In this example, d type functions are included considering the minimization of the electronic energy at the equilibrium geometry calculated at the Hartree-Fock level of theory. In the sequence, the analysis of the magnitude of the L C A O coefficients (weight functions) is carried out for the occupied molecular orbitals. The number of d type functions required to provide the minimum energy criterion changes from halogen to halogen. Iodine and bromine requires a set of four d type functions with ~o = - 1.68. Chlorine also requires four d type functions starting from 12o = - 0 . 9 6 , while fluorine uses three d type functions with 120 = - 0 . 9 6 . All the calculations use sets of 5d type functions (dxy, d ~ , dy:, d:z and d~2_,2). The results presented in Table 3, calculated with the universal basis set for the four dihalides, are in excellent agreement with available data from the literature calculated by other all-electron methods or with SBK pseudopotential and adapted basis func-
Table 3 Electronic properties o f the X 2 molecules ( G = equilibrium geometry, to~ = h a r m o n i c frequency and A E x = E x z - 2 E x ) in the ground state calculated with the pseudopotential and results from the literature Molecule
Method
r~/A
F2
HF a MP2 a MP4 ~ DFT h exp. ~
1.34 1.34 1.43 1.41 1.44
986.2 880.8 1040.0 917.0
- 1.67 1.55 1.54 2.62 1.72
HF ~ MP2 a M P 4 ~' DFT b AE J ECP ~ exp. c
1.99 1.99 2.02 2.10 2.06 2.05 1.99
591.3 535.4 494.0 541.1 528.2 565.0
0.38 1.92 2.19 2.21 2.30 2.18 2.48
HF a MP2" MP4 a AE d ECP e exp. ~
2.30 2.30 2.33 2.33 2.34 2.28
323.9 306.5 321.7 312.9 325.3
0.28 1.59 1.87 2.09 2.02 1.97
HF a MP2 a MP4 ~ DFT b exp. ~
2.68 2.69 2.71 2.65 2.67
220.0 209.0 221.0 215.0
0.49 1.68 1.69 2.17 1.54
CI 2
Br 2
12
o~e / c m -
I
A E x/eV
a This work. hCalculation with D F T using the pseudopotential and basis set of D Z V P quality plus nonlocal corrections o f Becke and Perdew
[30]. Ref. [31]. M C S C F calculation e m p l o y i n g T Z basis sets with all electrons
[32]. MCSCF calculation employingTZ basis sets with the pseudopotential of Hay and Wadt [32]. tions. The deviation can be associated either with the pseudopotential or with the original universal basis set. Larger universal basis set can be developed, yielding results in the limit of saturation of the basis sets [15]. The parameters for the basis set can be defined from the largest and smallest exponents of the universal basis set. A12 is then defined as: A12(k) = [12~(k) - 12o(k)]/(Nk - 1), where 12f(k) and 12o(k) are the logarithm of the largest and smallest exponents with symmetry k, and N k is the number of primitives in the mesh. N k can be large so as to saturate the universal basis set.
401
M. Giordan et al. / Chemical Physics Letters 279 (1997) 396-402
Table 4 Molecularproperties of the PbC12 molecule(r~ = equilibriumgeometry, 0 = equilibriumbond angle,/~ = dipole momentsand co= harmonic frequencies) in the ground state calculatedwith the universalbasis set and HW pseudopotentialand experimentalresults from the literature r¢/A
0/deg
/x/debye
~oI/cm- i
co~/cm t
o)~/cm- i
UB-MP2 ' UB-SCF ~' HW-SCF •
2,43 2.43 2,51
99.7 99,5 100.9
4.834 5.178 6.657
105.0 114.7 100.0
299.9 296.1 293.6
319.1 319.1 305.7
exp.
2.46
101:4
307 + 3 ~l 369 ~
329 +_3 '~ 365 ~
80 + I0 ~1 94 ~
MP2 calculationusing the universalbasis set with Hay and Wadt pseudopotential. b Hartree-Fock calculationusing the universalbasis set with Hay and Wadt pseudopotential. " Hartree-Fock calculationusing the Hay and Wadt pseudopotentialand the original basis set. d Ref. [33]. ~"Ref. [34].
A final application is carried out for PbCI 2, where relativistic effects are important to appropriately describe some molecular properties. The HW pseudopotential is used in these calculations, since the SBK pseudopotential does not take into account relativistic effects for heavier atoms. The s and p basis set for C1 is the same employed in the calculation of CI 2, discussed above. Three polarization functions were included with /2 o = - 0 . 9 6 . The basis set for Pb consists of the (1 l s l lp) universal mesh and two d polarization functions with 12o = - 3 . 8 4 , determined using the same criteria discussed for the diatomic halides. To minimize computational costs the adapted universal basis set for Pb and C1 were contracted by the D u n n i n g ' s segmented method [29]. The basis set for Pb was contracted from (1 lsl lp2d) to [5s5p2d] and the basis function for CI was contracted from ( l l s l l p 3 d ) to [9s9p3d]. The lost of energy after the basis set contraction was lower than 1 mE H with respect to the same calculation using the uncontracted basis set. Table 4 shows the results calculated with the universal basis set along with HW pseudopotential, with the HW pseudopotential and its original basis set and the experimental results from the literature. The calculated properties are in good agreement with experimental data. An interesting aspect can be observed when comparison is made with the results calculated via the original HW basis set and the universal basis set along with the HW pseudopotential. The C1-Pb-C1 bond angle calculated with the original HW basis set is closer to the experimental result than the universal basis set
data using the same pseudopotential. However, significant differences can be observed when other calculated molecular properties are compared. For instance, the dipole moment calculated using both basis set at the Hartree-Fock level of theory differs in almost 1.5 debye, a remarkable difference in the charge distribution. The absence of polarization functions and the small size of the original HW basis set can be pointed out as the responsible for the significant differences.
4. Conclusion In conclusion, the generator coordinate method proved to be adequate to adapt all-electron universal basis set to atoms represented by pseudopotentials. The use of the resulting universal basis set with atoms described by pseudopotentials yields results compatible with all-electron calculations. This paper indicates the possibility to expand or to reduce the size of the universal basis set either in the valence or in the inner region, using as criterion the analysis of the behavior of weight functions of the atomic or molecular systems. The polarization functions can also be easily obtained from the UB mesh with primitives of higher angular momentum through the analysis of the magnitude of the LCAO coefficients and some other molecular property of interest like the total electronic energy.
402
M. Giordan et a l . / Chemical Physics Letters 279 (1997) 396-402
Acknowledgements We acknowledge financial support from FAPESP (Funda~o de Ampfiro h Pesquisa do Estado de S~.o Paulo), CNPq (Conselho Nacional de Desenvolvimento Cientffico e Tecnol6gico) and CAPES (Coordenadoria de Aperfei~joamento de Pessoal de Ensino Superior). We also thank the computational facilities of Cenapad (Centro Nacional de Processamento de Alto Desempenho) - SP.
References [1] D.M. Silver, S. Wilson, W.C. Newport, Int. J. Quantum Chem. 14 (1978) 635. [2] D.M. Silver, W.C. Newport, Chem. Phys. Lett. 57 (1978) 421. [3] D.M. Silver, S. Wilson, J. Chem. Phys. 69 (1978) 3787. [4] S. Wilson, Adv. Chem. Phys. 67 (1987) 440. [5] D. Moncrieff, S. Wilson, J. Phys. B: At. Mol. Opt. Phys. 27 (1994) 1. [6] P.G. Mezey, Theor. Chim. Acta 53 (1979) 183. [7] J.R. Mohallem, Z. Phys. D: At. Mol. Clusters 3 (1986) 339. [8] J.R. Mohallem, M. Trsic, J, Chem. Phys. 86 (1987) 5043. [9] H.F.M. da Costa, M. Trsic, J.R. Mohallem, Mol. Phys. 62 (1987) 91. [10] A.B.F. da Silva, H.F.M. da Costa, M. Trsic, Mol. Phys. 68 (1989) 433. [11] R. Custodio, J.D. Goddard, M. Giordan, N.H. Morgon, Can. J. Chem. 70 (1992) 580. [12] G.L. Malli, A.B.F. da Silva, Y. lshikawa, Phys. Rev. 47 (1993) 143.
[13] R. Custodio, J.D. Goddard, J. Mol. Struct. (Theochem) 277 (1992) 263. [14] R. Custodio, M. Giordan. N.H. Morgon, J.D. Goddard, lnt. J. Quantum Chem. 42 (1992) 411. [15] R. Custodio, J.D. Goddard, J. Mol. Struct. (Theochem) 281 (1993) 75. [16] R. Custodio, W.M. Davis, J.D. Goddard, J, Mol. Struct. (Theochem) 315 (1994) 163. [17] N.H. Morgon, R. Custodio, J.M. Rivetos, Chem. Phys. Lett. 235 (1995) 436. [18] M. Giordan, R. Custodio, J. Comp. Chem. (in press). [19] R.D. Barrio. K. Ruedenberg, J. Chem. Phys. 59 (1973) 5936. [20] D.F, Feller, K. Ruedenberg, Theor. Chim. Acta 52 (1979) 231, [21] M.W. Schmidt, K. Ruedenberg, J. Chem. Phys. 71 (1979) 3951. [22] P.J. Hay, W.R. Wadt, J. Chem. Phys. 82 (1985) 270. [23] W.R. Wadt, P.J. Hay, J. Chem. Phys. 82 (1985) 284. [24] P.J. Hay, W.R. Wadt, J. Chem. Phys. 82 (1985) 299. [25] W. Stevens, H. Basch, J. Krauss. J. Chem, Phys. 81 (1984) 6026. [26] M. Dolg, U. Wedig, H. Stoll, H. Preuss, J, Chem. Phys. 66 (1987) 4377. [27] P.J. Hay, J. Chem. Phys. 66 (1977) 4377. [28] A.K. Rappe, T.A. Smedley, W.A. Goddard IlI, J. Phys. Chem. 85 (1981) 2607. [29] T.H. Dunning, J. Chem. Phys. 53 (1970) 2823. [30] H. Chert. M. Krasowski, G.J. Fitzgerald, J. Chem. Phys. 98 (1993) 8710. [31] K. Huber, G. Herzberg, Molecular Spectra and Molecular Structure IV, Constants of Diatomic Molecules, Van Nostrand Reinhold, New York, 1979. [32] M. Klobukovski, Theor. Chim. Acta 83 (1992) 239, [33] L. Brewer, G.R. Somayajulu, E. Brackett, Chem. Rev. 63 (1963) 111. [34] M. Dupuis, J. Rys, H.F. King, J. Chem. Phys. 65 (1976) 111.