- Email: [email protected]
Using fuzzy set theory to assess basis set quality
19 March 1999
Chemical Physics Letters 302 Ž1999. 273–280
Using fuzzy set theory to assess basis set quality J.A. Sordo
)
Departamento de Quımica Fısica y Analıtica, Facultad de Quımica, UniÕersidad de OÕiedo, Julian ´ ´ ´ ´ ´ ClaÕerıa ´ 8, 33006 OÕiedo, Principado de Asturias, Spain Received 28 July 1998; in final form 11 January 1999
Abstract An algorithm, based on Zadeh’s fuzzy set theory, to make an adequate basis set choice to compute simultaneously a number of molecular properties within Roothaan’s algebraic approximation is presented. Its usefulness is illustrated by analyzing fuzzy set predictions and actual results from molecular calculations on NiH 2 , a system for which it has been recently shown that the choice of an adequate basis set is important. q 1999 Elsevier Science B.V. All rights reserved.
1. Introduction As a direct consequence of Roothaan’s approximation to solve Hartree–Fock ŽHF. equations w1x the very first step in any computational project involving ab initio Žor density functional theory ŽDFT.. molecular calculations must be the choice of an appropriate basis set w2x. Although, most of the time, chemical intuition and previous experience are the tools commonly employed to make the choice, it is becoming more and more evident that basis sets, apparently of a rather similar quality according to these Žsubjective. criteria, can sometimes lead to quite different results 1. Therefore, the use of Žobjective. quality criteria available in the literature w4x should be strongly recommended. However, basis set quality is difficult to determine unambiguously. Indeed, it is well known that different quality tests make predictions that may not be coincident w5x. In a previous paper w6x, a rather simple mathematical formalism was developed to measure the degree of agreement between predictions from different quality tests. The basic ideas of the method can be summarized as follows. Let us define a quality test as w6x
Gf x 1 , x 2 , . . . , xn ; xr 4 s x i1 , x 2 , . . . , xn 4 ,
in gN , 1(in (n ,
Ž 1.
when < f Ž x i q . y f Ž xr . < - < f Ž x i p . y f Ž xr . < ,
Ž 2.
with ord x i p - ord x i q
where f is a mathematical function defining the quality test, x 1 , x 2 , . . . , xn 4 is a set of basis sets on which the quality test Gf is being applied, xr is a reference basis set, and ord x i p means the ordinal number ) 1
Corresponding author. Fax: q34 98 523 7850; e-mail: [email protected] See, as a representative illustration of this point Ref. w3x.
0009-2614r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 9 . 0 0 1 0 2 - 5
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corresponding to x i p g x i1 , x i2 , . . . , x i n 4 . If < f Ž x i q . y f Ž xr .< s < f Ž x i p . y f Ž xr .< then ord x i p - o r d x i q if x i q is of larger size than x i p . As a particular case, let f be a Žmolecular or atomic. property p Že.g., total energy, dipole moment, etc..
Gp x 1 , x 2 , . . . , xn ; xr 4 s x j1 , x j2 , . . . , x jn 4 ,
jn g N , 1 ( jn ( n .
Ž 3.
We may now define w6x
Gp x 1 , x 2 , . . . , xn ; xr 4 R Gf x 1 , x 2 , . . . , xn ; xr 4 s 1 y Ž qrqmax . ,
Ž 4.
where q is the least number of Žadjacent. transpositions needed to satisfy jn s i n ; n Ž q runs from 0 to qmax .. It has been shown w6x that if
Gp 1 x 1 , x 2 , . . . , xn ; xr 4 R Gf i x 1 , x 2 , . . . , xn ; xr 4 s q1 i , Gp 1 x 1 , x 2 , . . . , xn ; xr 4 R Gf j x 1 , x 2 , . . . , xn ; xr 4 s q1 j ,
Ž 5.
with q1 i ) q1 j , then the quality test Gf i is more appropriate than Gf j to assess basis set quality in order to compute property p 1. For a collection of properties p i Ž i s 1,2, . . . , P . and quality tests f j Ž j s 1,2, . . . , F . we can compute qi j s Gp i x 1 , x 2 , . . . , xn ; xr 4 R Gf j x 1 , x 2 , . . . , xn ; xr 4 .
Ž 6.
qi j measures how well the quality test f j predicts the correct ranking Žaccording to quality. of the basis sets considered in order to compute the property p i . The problem arises when, as generally occurs, one is interested in choosing an appropriate basis set to compute simultaneously two or more properties. The basis set requirements for calculations of different types of properties Ženergies, geometries, spectroscopic constants, electric properties, magnetic properties, . . . . are, in general, not alike w2x. None of the basis sets available are capable of describing accurately all the molecular properties that one may desire to calculate. The choice of an appropriate basis set must be done once the computational goals are well defined. Therefore, theoretical tools providing information on the basis set required in every particular case by considering the specific needs of the user are clearly needed. The use of an information theoretic approach w7x can be useful under such circumstances. In this Letter, we propose the use of a more powerful algorithm, based on the fuzzy set theory w8x, to make an adequate choice of basis set to develop a research project involving the calculation of several molecular properties.
2. Discussion Zadeh’s fuzzy set theory w8x provides the adequate theoretical background to deal with the vagueness inherent in subjective or imprecise determinations of preferences within the field of decision making. Bearing in mind the fuzzyness associated with the concept of basis set quality, we analyze in this Letter the way in which fuzzy set theory can be useful in order to choose a basis set. It is well known that a fuzzy set can be defined mathematically by assigning to each possible individual in the universe of discourse a value representing its grade of membership in the fuzzy set w8x. This grade corresponds to the degree to which that individual is similar or compatible with the concept represented by the fuzzy set. Thus, individuals may belong in the fuzzy set to a greater or lesser degree as indicated by a larger or smaller membership grade. These membership grades are very often represented by real number values ranking in the closed interval between 0 and 1. Let us define a fuzzy set Gp i associated with property p i , using conventional notation w9x, as Gp i s qi1rf 1 q qi2rf 2 q . . . qqi Frf F ,
Ž 7.
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where the membership grades are the values qi j Ž i s 1,2, . . . , P; j s 1,2, . . . , F . defined in Eq. Ž6.. It is clear that Gp i is a fuzzy representation of the appropriateness of different quality criteria f j Ž j s 1,2, . . . , F . to provide information about the basis set quality to compute property p i . According to Bellmann and Zadeh w10x, for a given collection of properties p 1 , p 2 , . . . , p P , the Gp i fuzzy sets combine to form a decision, D, which is a fuzzy set resulting from intersection of all Gp i , D s G p 1 l G p 2 l . . . l G pP s a 1rf 1 q a 2rf 2 q . . . qa Frf F ,
Ž 8.
a j s min w q1 j ,q2 j , . . . ,q P j x .
Ž 9.
with
The optimum Žcrisp. decision from D is that f j Žf opt . which maximizes the grade of membership to D,
a k s max Ž a 1 , a 2 , . . . , a F . ´ f opt s f k .
Ž 10 .
It has been tacitly assumed until now that all that properties in which we are interested are of equal importance. Bellman and Zadeh w10x have shown how the membership grades of the fuzzy set D might be expressed as a combination of the membership grades of the Gp i with the weighting coefficients reflecting the relative importance of the properties considered. The above ideas have been recently implemented in a computer program ŽQUALITY. which allows to analyze in detail basis set quality and to choose, in an automatic manner, the most appropriate basis set to compute one or more molecular properties accordingly w11x.
3. An illustrative example In a recent paper w3x Barron and co-workers reported calculations on the singlet 1A 1 and triplet 3D g states of NiH 2 using different methodologies and basis sets. Their results clearly show that the singlet–triplet energy separation and geometrical parameters ŽNi–H distances in the singlet and triplet states, and the H–Ni–H angle in the singlet state. are rather sensitive, in general, to the basis set composition. In order to apply the ideas presented in Section 2 we considered the following basis sets for nickel atom: Hehre’s Ž x 1 . w12x, Hay’s Ž x 2 . w13x, Clementi’s Ž x 3 . w14x, Huzinaga’s Ž x4 . w15x, Veillard’s Ž x 5 . w16x, Wachters–Hay’s Ž x6 . 2 , Huzinaga’s Ž x 7 . w17x, Huzinaga’s Ž x 8 . w18x, Veillard’s Ž x 9 . w16x, Wachters’ Ž x 10 . 3 , Hay’s Ž x 11 . w13x, Clementi’s Ž x 12 . w22x, and Wachters’ Ž x 13 . 4 . Clementi’s geometrical triple-zeta basis set Ž xr . w22x was taken as a reference basis set w11x. Details on the composition of these basis sets can be found in Table 1. A triple-zeta basis set 5 was used for hydrogen atoms in all cases Žexcept for the x 1 ŽSTO-3G. basis set.; thus, the difference in quality of the molecular calculations on NiH 2 can be mostly attributed to the nickel basis set composition. The properties we considered were: the singlet–triplet energy separation Žp 1 ., Ni–H distance for the triplet 3 D g state Žp 2 ., Ni–H distance for the singlet 1A 1 state Žp 3 ., and the H–Ni–H angle for the singlet 1A 1 state Žp 4 .. All molecular calculations were carried out using the GAUSSIAN 94 packages of programs w26x. The
2 This is the basis set used by GAUSSIAN 94 when 6-311G basis set is requested from the input for the nickel atom. See GAUSSIAN 94 User’s Reference manual for more details. 3 See Refs. w19–21x for contraction coefficients. 4 See Refs. w19–21x for contraction coefficients. 5 In the case of Wachters–Hay’s x6 basis set, the triple-zeta 6-311G basis set was used as in Ref. w3x Žsee Ref. w25x..
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J.A. Sordor Chemical Physics Letters 302 (1999) 273–280
Table 1 Description of the different basis sets considered Basis set xr Žreference.
Descriptiona Ž19r11r8.rw12r6r3x; Ž721111111111r511112r521.
x1 x2 x3 x4 x5 x6 x7 x8 x9 x 10 x 11 x 12 x 13
Ž12r6r3.rw4r2r1x; Ž3333r33r3. Ž12r6r5.rw5r2r2x; Ž62211r42r41. Ž15r8r5.rw5r2r2x; Ž82221r53r32. Ž13r7r4.rw5r3r2x; Ž43321r421r31. Ž12r6r4.rw8r4r2x; Ž42111111r3111r31. Ž14r9r5.rw9r5r3x; Ž611111111r51111r311. Ž14r9r5.rw14r9r5x Ž12r6r3.rw4r2r1x; Ž3333r33r3. Ž12r6r4.rw5r2r2x; Ž62211r42r31. Ž14r9r5.rw8r4r2x; Ž62111111r4212r32. Ž14r9r6.rw8r4r2x; Ž62111111r5112r51. Ž16r10r7.rw8r4r2x; Ž81111121r5113r43. Ž14r9r5.rw14r9r5x
a
The usual notation is used w23,24x.
quality criteria employed were: the traditional total energy test Ž E . and five overall information theoretic criteria ŽJ1–J5; see in Table 2 the quality tests considered in each case. w7x. Their selection was made bearing in mind previous studies on basis set quality of first-row transition metal elements w27,28x. The following strategy was applied w6x. Since the most time-consuming part of the algorithm presented in the previous sections is the calculation of Gp i x 1 , x 2 , . . . , xn ; xr 4 , which requires molecular calculations to be performed with all basis sets x 1 – xn 4 , we selected a subset of bases x 1 – x 7 4 Žsee Table 1. and by applying Eqs. Ž1. – Ž10. we determined the most appropriate quality test to assess basis set quality in order to compute properties p 1 –p 4 simultaneously at the HF level. Then we applied the quality test chosen to all basis sets x 1 – x 134 in order to get the final ranking of the basis sets considered in the present work. Table 3 collects the values of the properties p 1 –p 4 as computed at the HF level with basis sets x 1 – x 7 4 and Table 4 shows these basis sets ordered according to their quality as measured with the different quality tests Ž E, J1–J5. and properties Žp 1 –p 4 . considered. Program QUALITY w11x shows that according to Eq. Ž5., J3 is the most appropriate quality test to select the best basis set for computing properties p 1 and p 4 , J4 is the quality test chosen for p 2 and the total energy test, J1 and J2 criteria Žany of them. those selected for p 3 .
Table 2 Quality tests a used to construct the five overall information theoretic criteria ŽJ1–J5. employed Overall information
Quality tests
J1 J2 J3 J4 J5
E, ´ 3d , ² p : 3d , ² r : 3d E, ´ 4s , ² p :4 s , ² r :4 s E, DMAX, RDMAX, FWHMD, IMAX, RIMAX, FWHMI E, d J1jJ2jJ3jJ4
a
E, total energy; ´ , orbital energy; ² p :, momentum expectation value; ² r :, radial expectation value; DMAX, height of the radial density at RDMAX Žsee Ref. w27x.; RDMAX, peak position of the radial density; FWHMD, width of the radial density at RDMAX; IMAX, height of the radial orbital momentum density at RIMAX; RIMAX, peak position of the radial orbital momentum density; FWHMI, width of the radial orbital momentum density at RIMAX; d s dtest Žsee Ref. w28x..
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Table 3 ˚ ., Ni–H distance Žp 3 ; A˚ . for the singlet 1A 1 Singlet–triplet energy separations Žp 1; kcalrmol., Ni–H distance for the triplet 3D g state Žp 2 ; A 1 state, and H–Ni–H angle Žp 4 ; degrees. for the singlet A 1 state of NiH 2 as computed with different basis sets Ž a 1 – a 7 . at the HF level of theory Basis set
EŽ 1A 1 . – EŽ 3D g . Žp 1 .
Ni–HŽ 3D g . Žp 2 .
Ni–HŽ 1A 1 . Žp 3 .
H–Ni–HŽ 1A 1 . Žp 4 .
x1 x2 x3 x4 x5 x6 x7 xra
107.7 97.0 74.7 78.8 78.9 77.5 66.7 y5.7
1.642 1.566 1.616 1.595 1.593 1.592 1.523 1.543
1.550 1.572 1.533 1.532 1.531 1.530 1.458 1.427
98.3 179.9 112.6 122.8 123.0 118.3 103.9 87.8
a
Values computed at the DFT ŽB3LYP. level of theory are taken as a reference Žsee Ref. w11x..
From Eqs. Ž6. and Ž7. we obtain for the fuzzy sets associated with each property: Gp1 s 0.810rf 1 q 0.810rf 2 q 0.810rf 3 q 0.857rf 4 q 0.763rf5 q 0.810rf6 , Gp 2 s 0.810rf 1 q 0.810rf 2 q 0.810rf 3 q 0.762rf 4 q 0.857rf5 q 0.810rf6 ,
Ž 11 .
Gp 3 s 0.952rf 1 q 0.952rf 2 q 0.952rf 3 q 0.905rf 4 q 0.905rf5 q 0.952rf6 , Gp 4 s 0.524rf 1 q 0.524rf 2 q 0.524rf 3 q 0.571rf 4 q 0.476rf5 q 0.524rf6 , and consequently, from Eqs. Ž8. and Ž9., D s 0.524rf 1 q 0.524rf 2 q 0.524rf 3 q 0.571rf 4 q 0.476rf5 q 0.524rf6 .
Ž 12 .
According to Eq. Ž10. the optimum crisp decision from D is f 4 ŽJ3.; that is to say, J3 is the quality criterion that should be applied in order to select the best basis set to compute simultaneously p 1 , p 2 , p 3 and p 4 . Now we can apply J3 to ranking all basis sets x 1 – x 13 4 according to quality. Table 5 collects the results as obtained from program QUALITY w11x. In order to check the reliability of such predictions Žformulated from relatively inexpensive HF molecular calculations using a reduced number of basis sets. we carried out DFT ŽB3LYP. calculations w26x using a representative group of basis sets. Results are collected in Table 6.
Table 4 Basis sets ordered from lower Žleft. to higher Žright. quality according to the quality tests Ž E, J1–J5. and properties Žp 1 –p 4 . considered Quality test or property
Ranking
f 1 Ž E. f 2 ŽJ1. f 3 ŽJ2. f 4 ŽJ3. f 5 ŽJ4. f 6 ŽJ5. p1 p2 p3 p4
x1 x1 x1 x1 x1 x1 x1 x1 x2 x2
x2 x2 x2 x2 x3 x2 x2 x3 x1 x5
x3 x3 x3 x4 x2 x3 x5 x4 x3 x4
x4 x4 x4 x3 x4 x4 x4 x5 x4 x6
x5 x5 x5 x5 x5 x5 x6 x6 x5 x3
x6 x6 x6 x6 x6 x6 x3 x2 x6 x7
x7 x7 x7 x7 x7 x7 x7 x7 x7 x1
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Table 5 Basis sets ordered from lower Žtop. to higher Žbottom. quality according to the overall information theoretic measure J3 selected by fuzzy set theory Basis set
J3 a
x1 x8 x9 x2 x4 x3 x 12 x5 x 11 x 10 x6 x 13 x7
0.00 0.30 0.58 0.58 1.54 2.17 2.59 2.96 7.72 7.80 7.89 8.29 8.40
a
The values of J3 are computed by means of expressions derived in Ref. w7x Žthe greater the value of J3 the better the basis set quality..
Table 5 suggests a natural partition of basis sets according to quality: Ža. lower quality Ž x 1 , x 8 , x 9 , x 2 . Žb. intermediate quality Ž x4 , x 3 , x 12 x 5 . and Žc. higher quality Ž x 11 , x 10 , x6 , x 13 , x 7 .. Hay’s x 2 Ž55 primitive and 21 contracted basis functions. and Huzinaga’s x4 Ž55 primitive and 24 contracted basis functions. are rather similar in size. However, data in Table 6 show that Huzinaga’s basis set is clearly superior, as suggested by the J3 values for x 2 and x4 in Table 5, rendering values for the Ni–H distances of the singlet 1A 1 and triplet 3D g states closer to those obtained with xr . The singlet–triplet energy separation and the H–Ni–H angle are computed with a similar accuracy with x 2 and x4 basis sets. On the other hand, the rather similar quality of Hay’s x 11 and Wachters–Hay’s x6 basis sets predicted by the J3 values in Table 5 is fully confirmed by molecular calculations. Indeed, Table 6 shows that although x 11 basis set predicts the singlet 1A 1 to be lower in energy than the triplet 3D g Žin agreement with the most recent experimental studies w29x and high level ab initio calculations w3x., from a quantitative viewpoint, the singlet–triplet energy separation predicted by x 11 is, however, too large Žy21.7 kcalrmol vs. the best theoretical estimate of y5.7 kcalrmol.. On the contrary, x6 erroneously predicts the triplet 3D g to be lower in energy than the singlet 1A 1 but the singlet–triplet energy separation computed with this basis set Ž8.9 kcalrmol. is closer to the best theoretical estimate Žy5.7 kcalrmol.. The Ni–H distances for the singlet 1A 1 and triplet
Table 6 ˚ ., Ni–H distance Žp 3 ; A˚ . for the singlet 1A 1 Singlet–triplet energy separations Žp 1; kcalrmol., Ni–H distance for the triplet 3D g state Žp 2 ; A state, and H–Ni–H angle Žp 4 ; degrees. for the singlet 1A 1 state of NiH 2 as computed with selected basis sets at the DFT ŽB3LYP. level of theory Basis set
EŽ 1A 1 . – EŽ 3D g . Žp 1 .
Ni–HŽ 3D g . Žp 2 .
Ni–HŽ 1A 1 . Žp 3 .
H–Ni–HŽ 1A 1 . Žp 4 .
x2 x4 x 11 x6 x 13 x7 xr Žreference.
y25.2 14.6 y21.7 8.9 9.2 y3.6 y5.7
1.515 1.549 1.538 1.549 1.548 1.483 1.543
1.411 1.423 1.420 1.435 1.435 1.404 1.427
73.9 102.9 83.6 98.1 98.3 89.9 87.8
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3
D g states computed with both basis sets are quite similar, and the H–Ni–H angle computed with x 11 is slightly better. The rather similar quality predicted by fuzzy set theory for x6 and x 13 basis sets Žsee Table 5., despite of their very different size, is fully confirmed by molecular calculations Žsee Table 6.. The considerable smaller number of contracted basis functions of x6 Ž39 vs. 66 in the case of x 13 . notably reduces the computational cost. The high performance predicted for Huzinaga’s x 7 uncontracted basis set Žsee Table 5. is corroborated by the excellent values for the singlet–triplet energy separation, Ni–H distance for the singlet 1A 1 state and H–Ni–H angle rendered by this basis set, although the Ni–H distance for the triplet 3D g state is too short Žsee Table 6.. As a general conclusion, Wachters–Hay’s x6 basis set seems to be an excellent candidate to be used in ab initio or DFT calculations on NiH 2 . Once such a basis set is extended with appropriate diffuse functions Žspecially important in the present case given the well-known energetic proximity of several low-lying configurations in the nickel atom w28x., a high performance should be expected. Indeed, that is the case; DFT ŽB3LYP. calculations with x6 basis set extended with just one d diffuse function Žexponent 0.1316 taken from ˚ for the Ni–H distance of the Ref. w13x. give: y11.7 kcalrmol for the singlet–triplet energy separation, 1.540 A ˚ for the Ni–H distance of the singlet 1A 1 and 84.68 for the H–Ni–H angle of the singlet triplet 3D g state, 1.423 A 1 A 1. All these values are in good agreement with our best theoretical estimate Žsee xr in Table 6. and they were obtained with a remarkable reduction of the computational cost Ž71 primitive and 44 contracted basis functions for the extended x6 vs. 102 primitive and 98 contracted basis functions for xr ..
4. Conclusion It has been shown that Zadeh’s fuzzy set theory is an adequate tool in helping to make an appropriate basis set choice to carry out molecular calculations within Roothaan’s algebraic approximation. The Bellman–Zadeh algorithm has been applied to select the most appropriate basis set to compute a number of properties of NiH 2 that have been shown to be very sensitive to the basis set composition. Fuzzy set theory predictions agree quite well with the results obtained when the basis sets considered are used in molecular calculations to compute the above-mentioned properties.
References w1x C.C.J. Roothaan, Revs. Mod. Phys. 23 Ž1951. 69. w2x T. Helgaker, P.R. Taylor, in: D.R. de Yarkoni, Modern Electronic Structure Theory, Part II, Chap. 12, World Scientific, Singapore, 1995. w3x J.R. Barron, A.R. Kelley, R. Liu, J. Chem. Phys. 108 Ž1998. 1. w4x T.L. Sordo, C. Barrientos, J.A. Sordo, in: S. Fraga ŽEd.., Computational Chemistry: Structure, Interactions, and Reactivity, Part A, Chap. 9, Elsevier, Amsterdam, 1992. w5x R. Lopez, J.A. Sordo, T.L. Sordo, J. Mol. Struct. ŽTHEOCHEM. 253 Ž1995. 381. ´ w6x T.L. Sordo, J.A. Sordo, J. Mol. Struct. ŽTHEOCHEM. 253 Ž1992. 17. w7x G. Maroulis, Int. J. Quantum Chem. 34 Ž1988. 185. w8x L.A. Zadeh, Information Control 8 Ž1965. 338. w9x G.J. Klir, T.A. Folger, Fuzzy Sets, Uncertainty, and Information, Prentice Hall, Englewood Cliffs, NJ, 1988. w10x R.E. Bellman, L.A. Zadeh, Manag. Sci. 17 Ž1970. B141. w11x J.A. Sordo, Comput. Phys. Commun. 113 Ž1998. 85. w12x W.J. Pietro, W.J. Hehre, J. Comput. Chem. 4 Ž1983. 241. w13x P.J. Hay, J. Chem. Phys. 66 Ž1977. 4377.
280 w14x w15x w16x w17x w18x w19x w20x w21x w22x w23x w24x w25x w26x w27x w28x w29x
J.A. Sordor Chemical Physics Letters 302 (1999) 273–280 C. Urdaneta, G. Corongiu, E. Clementi, IBM Tech. Rep. KGN-133, Kingston, NY, 1987. H. Tatewaki, Y. Sakai, S. Huzinaga, J. Comput. Chem. 2 Ž1981. 278. B. Roos, A. Veillard, G. Vinot, Theoret. Chim. Acta ŽBerl.. 20 Ž1971. 1. S. Huzinaga, J. Chem. Phys. 66 Ž1977. 4245. S. Huzinaga ŽEd.., Gaussian Basis Sets for Molecular Calculations, Elsevier, Amsterdam, 1984. A.J.H. Wachters, J. Chem. Phys. 52 Ž1970. 1033. A.J.H. Wachters, IBM Tech. Rep. RJ 584, 1969. R. Poirier, R. Kari, I.G. Csizmadia, Handbook of Gaussian Basis Sets, Elsevier, Amsterdam, 1985. R. Gomperts, E. Clementi, IBM Tech. Rep. KGN-118, Kingston, NY, 1987. T.H. Dunning, Jr. P.J. Hay, in: H.F. Schaefer III ŽEd.., Methods of Electronic Structure Theory, vol. 3, Plenum Press, New York, 1977. S. Huzinaga, Comput. Phys. Rep. 2 Ž1985. 279 T.H. Dunning, J. Chem. Phys. 55 Ž1971. 716. M.J. Frisch et al., GAUSSIAN 94, Gaussian, Pittsburgh, PA, 1995. P.E. Regier, J. Fischer, B.S. Sharma, A.J. Thakkar, Int. J. Quantum Chem. 28 Ž1985. 429. L.A.G. Bernardo, J.A. Sordo, J. Chem. Phys. 85 Ž1986. 1475. S. Li, R.J. van Zee, W. Weltner Jr., M.G. Cory, M.C. Zerner, J. Chem. Phys. 106 Ž1997. 2055.
Chemical Physics Letters 302 Ž1999. 273–280
Using fuzzy set theory to assess basis set quality J.A. Sordo
)
Departamento de Quımica Fısica y Analıtica, Facultad de Quımica, UniÕersidad de OÕiedo, Julian ´ ´ ´ ´ ´ ClaÕerıa ´ 8, 33006 OÕiedo, Principado de Asturias, Spain Received 28 July 1998; in final form 11 January 1999
Abstract An algorithm, based on Zadeh’s fuzzy set theory, to make an adequate basis set choice to compute simultaneously a number of molecular properties within Roothaan’s algebraic approximation is presented. Its usefulness is illustrated by analyzing fuzzy set predictions and actual results from molecular calculations on NiH 2 , a system for which it has been recently shown that the choice of an adequate basis set is important. q 1999 Elsevier Science B.V. All rights reserved.
1. Introduction As a direct consequence of Roothaan’s approximation to solve Hartree–Fock ŽHF. equations w1x the very first step in any computational project involving ab initio Žor density functional theory ŽDFT.. molecular calculations must be the choice of an appropriate basis set w2x. Although, most of the time, chemical intuition and previous experience are the tools commonly employed to make the choice, it is becoming more and more evident that basis sets, apparently of a rather similar quality according to these Žsubjective. criteria, can sometimes lead to quite different results 1. Therefore, the use of Žobjective. quality criteria available in the literature w4x should be strongly recommended. However, basis set quality is difficult to determine unambiguously. Indeed, it is well known that different quality tests make predictions that may not be coincident w5x. In a previous paper w6x, a rather simple mathematical formalism was developed to measure the degree of agreement between predictions from different quality tests. The basic ideas of the method can be summarized as follows. Let us define a quality test as w6x
Gf x 1 , x 2 , . . . , xn ; xr 4 s x i1 , x 2 , . . . , xn 4 ,
in gN , 1(in (n ,
Ž 1.
when < f Ž x i q . y f Ž xr . < - < f Ž x i p . y f Ž xr . < ,
Ž 2.
with ord x i p - ord x i q
where f is a mathematical function defining the quality test, x 1 , x 2 , . . . , xn 4 is a set of basis sets on which the quality test Gf is being applied, xr is a reference basis set, and ord x i p means the ordinal number ) 1
Corresponding author. Fax: q34 98 523 7850; e-mail: [email protected] See, as a representative illustration of this point Ref. w3x.
0009-2614r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 9 . 0 0 1 0 2 - 5
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corresponding to x i p g x i1 , x i2 , . . . , x i n 4 . If < f Ž x i q . y f Ž xr .< s < f Ž x i p . y f Ž xr .< then ord x i p - o r d x i q if x i q is of larger size than x i p . As a particular case, let f be a Žmolecular or atomic. property p Že.g., total energy, dipole moment, etc..
Gp x 1 , x 2 , . . . , xn ; xr 4 s x j1 , x j2 , . . . , x jn 4 ,
jn g N , 1 ( jn ( n .
Ž 3.
We may now define w6x
Gp x 1 , x 2 , . . . , xn ; xr 4 R Gf x 1 , x 2 , . . . , xn ; xr 4 s 1 y Ž qrqmax . ,
Ž 4.
where q is the least number of Žadjacent. transpositions needed to satisfy jn s i n ; n Ž q runs from 0 to qmax .. It has been shown w6x that if
Gp 1 x 1 , x 2 , . . . , xn ; xr 4 R Gf i x 1 , x 2 , . . . , xn ; xr 4 s q1 i , Gp 1 x 1 , x 2 , . . . , xn ; xr 4 R Gf j x 1 , x 2 , . . . , xn ; xr 4 s q1 j ,
Ž 5.
with q1 i ) q1 j , then the quality test Gf i is more appropriate than Gf j to assess basis set quality in order to compute property p 1. For a collection of properties p i Ž i s 1,2, . . . , P . and quality tests f j Ž j s 1,2, . . . , F . we can compute qi j s Gp i x 1 , x 2 , . . . , xn ; xr 4 R Gf j x 1 , x 2 , . . . , xn ; xr 4 .
Ž 6.
qi j measures how well the quality test f j predicts the correct ranking Žaccording to quality. of the basis sets considered in order to compute the property p i . The problem arises when, as generally occurs, one is interested in choosing an appropriate basis set to compute simultaneously two or more properties. The basis set requirements for calculations of different types of properties Ženergies, geometries, spectroscopic constants, electric properties, magnetic properties, . . . . are, in general, not alike w2x. None of the basis sets available are capable of describing accurately all the molecular properties that one may desire to calculate. The choice of an appropriate basis set must be done once the computational goals are well defined. Therefore, theoretical tools providing information on the basis set required in every particular case by considering the specific needs of the user are clearly needed. The use of an information theoretic approach w7x can be useful under such circumstances. In this Letter, we propose the use of a more powerful algorithm, based on the fuzzy set theory w8x, to make an adequate choice of basis set to develop a research project involving the calculation of several molecular properties.
2. Discussion Zadeh’s fuzzy set theory w8x provides the adequate theoretical background to deal with the vagueness inherent in subjective or imprecise determinations of preferences within the field of decision making. Bearing in mind the fuzzyness associated with the concept of basis set quality, we analyze in this Letter the way in which fuzzy set theory can be useful in order to choose a basis set. It is well known that a fuzzy set can be defined mathematically by assigning to each possible individual in the universe of discourse a value representing its grade of membership in the fuzzy set w8x. This grade corresponds to the degree to which that individual is similar or compatible with the concept represented by the fuzzy set. Thus, individuals may belong in the fuzzy set to a greater or lesser degree as indicated by a larger or smaller membership grade. These membership grades are very often represented by real number values ranking in the closed interval between 0 and 1. Let us define a fuzzy set Gp i associated with property p i , using conventional notation w9x, as Gp i s qi1rf 1 q qi2rf 2 q . . . qqi Frf F ,
Ž 7.
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where the membership grades are the values qi j Ž i s 1,2, . . . , P; j s 1,2, . . . , F . defined in Eq. Ž6.. It is clear that Gp i is a fuzzy representation of the appropriateness of different quality criteria f j Ž j s 1,2, . . . , F . to provide information about the basis set quality to compute property p i . According to Bellmann and Zadeh w10x, for a given collection of properties p 1 , p 2 , . . . , p P , the Gp i fuzzy sets combine to form a decision, D, which is a fuzzy set resulting from intersection of all Gp i , D s G p 1 l G p 2 l . . . l G pP s a 1rf 1 q a 2rf 2 q . . . qa Frf F ,
Ž 8.
a j s min w q1 j ,q2 j , . . . ,q P j x .
Ž 9.
with
The optimum Žcrisp. decision from D is that f j Žf opt . which maximizes the grade of membership to D,
a k s max Ž a 1 , a 2 , . . . , a F . ´ f opt s f k .
Ž 10 .
It has been tacitly assumed until now that all that properties in which we are interested are of equal importance. Bellman and Zadeh w10x have shown how the membership grades of the fuzzy set D might be expressed as a combination of the membership grades of the Gp i with the weighting coefficients reflecting the relative importance of the properties considered. The above ideas have been recently implemented in a computer program ŽQUALITY. which allows to analyze in detail basis set quality and to choose, in an automatic manner, the most appropriate basis set to compute one or more molecular properties accordingly w11x.
3. An illustrative example In a recent paper w3x Barron and co-workers reported calculations on the singlet 1A 1 and triplet 3D g states of NiH 2 using different methodologies and basis sets. Their results clearly show that the singlet–triplet energy separation and geometrical parameters ŽNi–H distances in the singlet and triplet states, and the H–Ni–H angle in the singlet state. are rather sensitive, in general, to the basis set composition. In order to apply the ideas presented in Section 2 we considered the following basis sets for nickel atom: Hehre’s Ž x 1 . w12x, Hay’s Ž x 2 . w13x, Clementi’s Ž x 3 . w14x, Huzinaga’s Ž x4 . w15x, Veillard’s Ž x 5 . w16x, Wachters–Hay’s Ž x6 . 2 , Huzinaga’s Ž x 7 . w17x, Huzinaga’s Ž x 8 . w18x, Veillard’s Ž x 9 . w16x, Wachters’ Ž x 10 . 3 , Hay’s Ž x 11 . w13x, Clementi’s Ž x 12 . w22x, and Wachters’ Ž x 13 . 4 . Clementi’s geometrical triple-zeta basis set Ž xr . w22x was taken as a reference basis set w11x. Details on the composition of these basis sets can be found in Table 1. A triple-zeta basis set 5 was used for hydrogen atoms in all cases Žexcept for the x 1 ŽSTO-3G. basis set.; thus, the difference in quality of the molecular calculations on NiH 2 can be mostly attributed to the nickel basis set composition. The properties we considered were: the singlet–triplet energy separation Žp 1 ., Ni–H distance for the triplet 3 D g state Žp 2 ., Ni–H distance for the singlet 1A 1 state Žp 3 ., and the H–Ni–H angle for the singlet 1A 1 state Žp 4 .. All molecular calculations were carried out using the GAUSSIAN 94 packages of programs w26x. The
2 This is the basis set used by GAUSSIAN 94 when 6-311G basis set is requested from the input for the nickel atom. See GAUSSIAN 94 User’s Reference manual for more details. 3 See Refs. w19–21x for contraction coefficients. 4 See Refs. w19–21x for contraction coefficients. 5 In the case of Wachters–Hay’s x6 basis set, the triple-zeta 6-311G basis set was used as in Ref. w3x Žsee Ref. w25x..
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Table 1 Description of the different basis sets considered Basis set xr Žreference.
Descriptiona Ž19r11r8.rw12r6r3x; Ž721111111111r511112r521.
x1 x2 x3 x4 x5 x6 x7 x8 x9 x 10 x 11 x 12 x 13
Ž12r6r3.rw4r2r1x; Ž3333r33r3. Ž12r6r5.rw5r2r2x; Ž62211r42r41. Ž15r8r5.rw5r2r2x; Ž82221r53r32. Ž13r7r4.rw5r3r2x; Ž43321r421r31. Ž12r6r4.rw8r4r2x; Ž42111111r3111r31. Ž14r9r5.rw9r5r3x; Ž611111111r51111r311. Ž14r9r5.rw14r9r5x Ž12r6r3.rw4r2r1x; Ž3333r33r3. Ž12r6r4.rw5r2r2x; Ž62211r42r31. Ž14r9r5.rw8r4r2x; Ž62111111r4212r32. Ž14r9r6.rw8r4r2x; Ž62111111r5112r51. Ž16r10r7.rw8r4r2x; Ž81111121r5113r43. Ž14r9r5.rw14r9r5x
a
The usual notation is used w23,24x.
quality criteria employed were: the traditional total energy test Ž E . and five overall information theoretic criteria ŽJ1–J5; see in Table 2 the quality tests considered in each case. w7x. Their selection was made bearing in mind previous studies on basis set quality of first-row transition metal elements w27,28x. The following strategy was applied w6x. Since the most time-consuming part of the algorithm presented in the previous sections is the calculation of Gp i x 1 , x 2 , . . . , xn ; xr 4 , which requires molecular calculations to be performed with all basis sets x 1 – xn 4 , we selected a subset of bases x 1 – x 7 4 Žsee Table 1. and by applying Eqs. Ž1. – Ž10. we determined the most appropriate quality test to assess basis set quality in order to compute properties p 1 –p 4 simultaneously at the HF level. Then we applied the quality test chosen to all basis sets x 1 – x 134 in order to get the final ranking of the basis sets considered in the present work. Table 3 collects the values of the properties p 1 –p 4 as computed at the HF level with basis sets x 1 – x 7 4 and Table 4 shows these basis sets ordered according to their quality as measured with the different quality tests Ž E, J1–J5. and properties Žp 1 –p 4 . considered. Program QUALITY w11x shows that according to Eq. Ž5., J3 is the most appropriate quality test to select the best basis set for computing properties p 1 and p 4 , J4 is the quality test chosen for p 2 and the total energy test, J1 and J2 criteria Žany of them. those selected for p 3 .
Table 2 Quality tests a used to construct the five overall information theoretic criteria ŽJ1–J5. employed Overall information
Quality tests
J1 J2 J3 J4 J5
E, ´ 3d , ² p : 3d , ² r : 3d E, ´ 4s , ² p :4 s , ² r :4 s E, DMAX, RDMAX, FWHMD, IMAX, RIMAX, FWHMI E, d J1jJ2jJ3jJ4
a
E, total energy; ´ , orbital energy; ² p :, momentum expectation value; ² r :, radial expectation value; DMAX, height of the radial density at RDMAX Žsee Ref. w27x.; RDMAX, peak position of the radial density; FWHMD, width of the radial density at RDMAX; IMAX, height of the radial orbital momentum density at RIMAX; RIMAX, peak position of the radial orbital momentum density; FWHMI, width of the radial orbital momentum density at RIMAX; d s dtest Žsee Ref. w28x..
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Table 3 ˚ ., Ni–H distance Žp 3 ; A˚ . for the singlet 1A 1 Singlet–triplet energy separations Žp 1; kcalrmol., Ni–H distance for the triplet 3D g state Žp 2 ; A 1 state, and H–Ni–H angle Žp 4 ; degrees. for the singlet A 1 state of NiH 2 as computed with different basis sets Ž a 1 – a 7 . at the HF level of theory Basis set
EŽ 1A 1 . – EŽ 3D g . Žp 1 .
Ni–HŽ 3D g . Žp 2 .
Ni–HŽ 1A 1 . Žp 3 .
H–Ni–HŽ 1A 1 . Žp 4 .
x1 x2 x3 x4 x5 x6 x7 xra
107.7 97.0 74.7 78.8 78.9 77.5 66.7 y5.7
1.642 1.566 1.616 1.595 1.593 1.592 1.523 1.543
1.550 1.572 1.533 1.532 1.531 1.530 1.458 1.427
98.3 179.9 112.6 122.8 123.0 118.3 103.9 87.8
a
Values computed at the DFT ŽB3LYP. level of theory are taken as a reference Žsee Ref. w11x..
From Eqs. Ž6. and Ž7. we obtain for the fuzzy sets associated with each property: Gp1 s 0.810rf 1 q 0.810rf 2 q 0.810rf 3 q 0.857rf 4 q 0.763rf5 q 0.810rf6 , Gp 2 s 0.810rf 1 q 0.810rf 2 q 0.810rf 3 q 0.762rf 4 q 0.857rf5 q 0.810rf6 ,
Ž 11 .
Gp 3 s 0.952rf 1 q 0.952rf 2 q 0.952rf 3 q 0.905rf 4 q 0.905rf5 q 0.952rf6 , Gp 4 s 0.524rf 1 q 0.524rf 2 q 0.524rf 3 q 0.571rf 4 q 0.476rf5 q 0.524rf6 , and consequently, from Eqs. Ž8. and Ž9., D s 0.524rf 1 q 0.524rf 2 q 0.524rf 3 q 0.571rf 4 q 0.476rf5 q 0.524rf6 .
Ž 12 .
According to Eq. Ž10. the optimum crisp decision from D is f 4 ŽJ3.; that is to say, J3 is the quality criterion that should be applied in order to select the best basis set to compute simultaneously p 1 , p 2 , p 3 and p 4 . Now we can apply J3 to ranking all basis sets x 1 – x 13 4 according to quality. Table 5 collects the results as obtained from program QUALITY w11x. In order to check the reliability of such predictions Žformulated from relatively inexpensive HF molecular calculations using a reduced number of basis sets. we carried out DFT ŽB3LYP. calculations w26x using a representative group of basis sets. Results are collected in Table 6.
Table 4 Basis sets ordered from lower Žleft. to higher Žright. quality according to the quality tests Ž E, J1–J5. and properties Žp 1 –p 4 . considered Quality test or property
Ranking
f 1 Ž E. f 2 ŽJ1. f 3 ŽJ2. f 4 ŽJ3. f 5 ŽJ4. f 6 ŽJ5. p1 p2 p3 p4
x1 x1 x1 x1 x1 x1 x1 x1 x2 x2
x2 x2 x2 x2 x3 x2 x2 x3 x1 x5
x3 x3 x3 x4 x2 x3 x5 x4 x3 x4
x4 x4 x4 x3 x4 x4 x4 x5 x4 x6
x5 x5 x5 x5 x5 x5 x6 x6 x5 x3
x6 x6 x6 x6 x6 x6 x3 x2 x6 x7
x7 x7 x7 x7 x7 x7 x7 x7 x7 x1
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Table 5 Basis sets ordered from lower Žtop. to higher Žbottom. quality according to the overall information theoretic measure J3 selected by fuzzy set theory Basis set
J3 a
x1 x8 x9 x2 x4 x3 x 12 x5 x 11 x 10 x6 x 13 x7
0.00 0.30 0.58 0.58 1.54 2.17 2.59 2.96 7.72 7.80 7.89 8.29 8.40
a
The values of J3 are computed by means of expressions derived in Ref. w7x Žthe greater the value of J3 the better the basis set quality..
Table 5 suggests a natural partition of basis sets according to quality: Ža. lower quality Ž x 1 , x 8 , x 9 , x 2 . Žb. intermediate quality Ž x4 , x 3 , x 12 x 5 . and Žc. higher quality Ž x 11 , x 10 , x6 , x 13 , x 7 .. Hay’s x 2 Ž55 primitive and 21 contracted basis functions. and Huzinaga’s x4 Ž55 primitive and 24 contracted basis functions. are rather similar in size. However, data in Table 6 show that Huzinaga’s basis set is clearly superior, as suggested by the J3 values for x 2 and x4 in Table 5, rendering values for the Ni–H distances of the singlet 1A 1 and triplet 3D g states closer to those obtained with xr . The singlet–triplet energy separation and the H–Ni–H angle are computed with a similar accuracy with x 2 and x4 basis sets. On the other hand, the rather similar quality of Hay’s x 11 and Wachters–Hay’s x6 basis sets predicted by the J3 values in Table 5 is fully confirmed by molecular calculations. Indeed, Table 6 shows that although x 11 basis set predicts the singlet 1A 1 to be lower in energy than the triplet 3D g Žin agreement with the most recent experimental studies w29x and high level ab initio calculations w3x., from a quantitative viewpoint, the singlet–triplet energy separation predicted by x 11 is, however, too large Žy21.7 kcalrmol vs. the best theoretical estimate of y5.7 kcalrmol.. On the contrary, x6 erroneously predicts the triplet 3D g to be lower in energy than the singlet 1A 1 but the singlet–triplet energy separation computed with this basis set Ž8.9 kcalrmol. is closer to the best theoretical estimate Žy5.7 kcalrmol.. The Ni–H distances for the singlet 1A 1 and triplet
Table 6 ˚ ., Ni–H distance Žp 3 ; A˚ . for the singlet 1A 1 Singlet–triplet energy separations Žp 1; kcalrmol., Ni–H distance for the triplet 3D g state Žp 2 ; A state, and H–Ni–H angle Žp 4 ; degrees. for the singlet 1A 1 state of NiH 2 as computed with selected basis sets at the DFT ŽB3LYP. level of theory Basis set
EŽ 1A 1 . – EŽ 3D g . Žp 1 .
Ni–HŽ 3D g . Žp 2 .
Ni–HŽ 1A 1 . Žp 3 .
H–Ni–HŽ 1A 1 . Žp 4 .
x2 x4 x 11 x6 x 13 x7 xr Žreference.
y25.2 14.6 y21.7 8.9 9.2 y3.6 y5.7
1.515 1.549 1.538 1.549 1.548 1.483 1.543
1.411 1.423 1.420 1.435 1.435 1.404 1.427
73.9 102.9 83.6 98.1 98.3 89.9 87.8
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3
D g states computed with both basis sets are quite similar, and the H–Ni–H angle computed with x 11 is slightly better. The rather similar quality predicted by fuzzy set theory for x6 and x 13 basis sets Žsee Table 5., despite of their very different size, is fully confirmed by molecular calculations Žsee Table 6.. The considerable smaller number of contracted basis functions of x6 Ž39 vs. 66 in the case of x 13 . notably reduces the computational cost. The high performance predicted for Huzinaga’s x 7 uncontracted basis set Žsee Table 5. is corroborated by the excellent values for the singlet–triplet energy separation, Ni–H distance for the singlet 1A 1 state and H–Ni–H angle rendered by this basis set, although the Ni–H distance for the triplet 3D g state is too short Žsee Table 6.. As a general conclusion, Wachters–Hay’s x6 basis set seems to be an excellent candidate to be used in ab initio or DFT calculations on NiH 2 . Once such a basis set is extended with appropriate diffuse functions Žspecially important in the present case given the well-known energetic proximity of several low-lying configurations in the nickel atom w28x., a high performance should be expected. Indeed, that is the case; DFT ŽB3LYP. calculations with x6 basis set extended with just one d diffuse function Žexponent 0.1316 taken from ˚ for the Ni–H distance of the Ref. w13x. give: y11.7 kcalrmol for the singlet–triplet energy separation, 1.540 A ˚ for the Ni–H distance of the singlet 1A 1 and 84.68 for the H–Ni–H angle of the singlet triplet 3D g state, 1.423 A 1 A 1. All these values are in good agreement with our best theoretical estimate Žsee xr in Table 6. and they were obtained with a remarkable reduction of the computational cost Ž71 primitive and 44 contracted basis functions for the extended x6 vs. 102 primitive and 98 contracted basis functions for xr ..
4. Conclusion It has been shown that Zadeh’s fuzzy set theory is an adequate tool in helping to make an appropriate basis set choice to carry out molecular calculations within Roothaan’s algebraic approximation. The Bellman–Zadeh algorithm has been applied to select the most appropriate basis set to compute a number of properties of NiH 2 that have been shown to be very sensitive to the basis set composition. Fuzzy set theory predictions agree quite well with the results obtained when the basis sets considered are used in molecular calculations to compute the above-mentioned properties.
References w1x C.C.J. Roothaan, Revs. Mod. Phys. 23 Ž1951. 69. w2x T. Helgaker, P.R. Taylor, in: D.R. de Yarkoni, Modern Electronic Structure Theory, Part II, Chap. 12, World Scientific, Singapore, 1995. w3x J.R. Barron, A.R. Kelley, R. Liu, J. Chem. Phys. 108 Ž1998. 1. w4x T.L. Sordo, C. Barrientos, J.A. Sordo, in: S. Fraga ŽEd.., Computational Chemistry: Structure, Interactions, and Reactivity, Part A, Chap. 9, Elsevier, Amsterdam, 1992. w5x R. Lopez, J.A. Sordo, T.L. Sordo, J. Mol. Struct. ŽTHEOCHEM. 253 Ž1995. 381. ´ w6x T.L. Sordo, J.A. Sordo, J. Mol. Struct. ŽTHEOCHEM. 253 Ž1992. 17. w7x G. Maroulis, Int. J. Quantum Chem. 34 Ž1988. 185. w8x L.A. Zadeh, Information Control 8 Ž1965. 338. w9x G.J. Klir, T.A. Folger, Fuzzy Sets, Uncertainty, and Information, Prentice Hall, Englewood Cliffs, NJ, 1988. w10x R.E. Bellman, L.A. Zadeh, Manag. Sci. 17 Ž1970. B141. w11x J.A. Sordo, Comput. Phys. Commun. 113 Ž1998. 85. w12x W.J. Pietro, W.J. Hehre, J. Comput. Chem. 4 Ž1983. 241. w13x P.J. Hay, J. Chem. Phys. 66 Ž1977. 4377.
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