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Accurate universal basis set for H through Xe for Hartree–Fock calculations
Chemical Physics 233 Ž1998. 1–7
Accurate universal basis set for H through Xe for Hartree–Fock calculations F.E. Jorge ) , R.F. Martins Departamento de Fısica — CCE, UniÕersidade Federal do Espırito ´ ´ Santo, 29060-900 Vitoria, ´ ES, Brazil Received 22 July 1997
Abstract We have applied the generator coordinate Hartree–Fock method to generate Slater-type functions for the atoms from H through Xe. The Griffin–Hill–Wheeler–Hartree–Fock equations are integrated numerically generating a universal basis set for these atoms. When compared with the values obtained by Koga et al. wPhys. Rev. A 47 Ž1993. 4510x using a smaller but fully-optimized basis sets, it is generally observed that our Hartree–Fock ground state energies are equal for the atoms from He Ž Z s 2. to Mg Ž Z s 12., larger for the atoms from Al Ž Z s 13. to Kr Ž Z s 36., and lower for the atoms from Rb Ž Z s 37. to Xe Ž Z s 54.. Moreover, our energy results compare satisfactorily with the corresponding atom-optimized and numerical Hartree–Fock calculations of Koga et al. wJ. Chem. Phys. 103 Ž1995. 3000x. q 1998 Elsevier Science B.V. All rights reserved.
1. Introduction In 1986 Mohallen et al. w1x introduced a generator coordinate version of the Hartree–Fock ŽGCHF. method. One of the first applications of the GCHF method was the generation of nonrelativistic Gaussian- and Slater-type functions ŽGTF an STF, respectively. for the atoms of the first and second rows of the Periodic Table w2,3x. Later, da Silva and Trsic presented GTF and STF universal basis sets ŽUBS. encompassing the atoms from H through Xe for atomic and molecular calculations w4x, and recently the GCHF method was tested successfully in the generation of large universal Gaussian basis sets ŽUGBSs. for the atoms H Ž Z s 1. through La Ž Z s 57. w5x and for the heavy atoms from Ce Ž Z s 58. through Lr Ž Z s 103. w6x. We recall that the UGBSs )
Corresponding author. E-mail: [email protected]
w5,6x can be easily used as starting basis sets in relativistic calculations Žsee Ref. w7x.. The first unique set of exponents for various different atoms was found with the even tempered STF basis by Silver et al. w8,9x; applications of universal atomic basis sets to molecular systems were initiated by Cooper and Wilson w10,11x. Another practical application of universal atomic basis sets could be the storage of an unique set of integrals to be used in configuration interaction or perturbation Žinternal or external. calculations for different atoms. Hartree–Fock ŽHF. calculations consider the nuclei represented as particle points. In this case, STF have the correct functional form to describe the nonrelativistic wave function of the atoms near the origin, but they are not particularly suitable for analytic self-consistent field ŽSCF. molecular calculations. On the other hand, GTF are useful in the
0301-0104r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 1 - 0 1 0 4 Ž 9 8 . 0 0 1 3 6 - 0
F.E. Jorge, R.F. Martinsr Chemical Physics 233 (1998) 1–7
2
Table 1 Discretization parametersa of the universal basis set ŽUBS. Z
Atom
N
V minŽs.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51
H He Li Be B C N O F Ne Na Mg Al Si P S Cl Ar K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb
7s 8s 12s 11s 10s6p 12s7p 11s8p 12s8p 12s8p 11s8p 12s8p 12s8p 12s9p 12s9p 12s9p 12s9p 12s9p 12s10p 12s10p 12s10p 13s10p8d 13s9p7d 13s10p7d 12s10p7d 12s10p8d 12s10p8d 13s10p7d 13s10p8d 13s10p8d 13s10p8d 13s11p7d 13s11p8d 13s10p8d 13s10p8d 13s10p9d 13s10p9d 13s10p9d 13s9p7d 13s10p9d 14s10p9d 14s10p9d 14s10p9d 14s10p9d 14s10p9d 14s10p9d 14s10p9d 14s10p9d 14s10p9d 13s12p9d 13s11p9d 13s11p9d
y0.1330 y0.1330 y0.3319 y0.1993 y0.0667 y0.1330 y0.0667 y0.0667 y0.0667 y0.0004 y0.1993 -0.0667 y0.0667 y0.0667 y0.0667 y0.0667 y0.0667 y0.0004 y0.1330 y0.1330 y0.1330 y0.1330 y0.1330 y0.1330 y0.0667 y0.0667 y0.0667 y0.0667 y0.0667 y0.0667 y0.0667 y0.0667 y0.0004 y0.0004 y0.0004 y0.0004 y0.1330 y0.1330 y0.0667 y0.0667 y0.0667 y0.0667 y0.0667 y0.0667 y0.0667 y0.0004 y0.0667 y0.0667 y0.0667 y0.0004 y0.0004
V minŽp.
V minŽd.
Table 1 Žcontinued. Z
Atom
N
V minŽs.
V minŽp.
V minŽd.
52 53 54
Te I Xe
13s11p8d 13s11p9d 13s11p9d
y0.0004 y0.0004 y0.0004
y0.0004 y0.0004 y0.0004
0.1322 0.1322 0.0659
a
y0.0667 y0.0667 y0.0667 y0.0667 y0.0667 y0.0004 0.0659 y0.0004 y0.0667 y0.0667 y0.0004 y0.0004 y0.0667 y0.0004 y0.0004 y0.0004 y0.0004 y0.0004 y0.0667 y0.0004 0.0659 0.0659 0.1322 0.1322 0.1322 0.0659 y0.0667 y0.0667 y0.0004 y0.0004 y0.0004 y0.0004 y0.0004 0.0659 0.0659 0.0659 0.0659 0.0659 0.1322 0.1322 0.1322 0.1322 0.1322 0.1322 y0.0667 y0.0004 y0.0004
The increment value Žsee Eq. Ž4.. used for all symmetries of all atoms ŽH to Xe. is equal to D V s 0.0663.
evaluation of multi-center integrals in molecules but they do not possess the correct functional behavior near the origin. In this work we present a highly accurate universal STF basis set for H to Xe generated with the GCHF method. 2. Computational procedures
y0.0004 y0.0004 y0.0004 y0.0004 0.0659 0.0659 0.0659 y0.0004 y0.0004 0.0659 0.0659 0.1322 0.1322 0.1322 0.1322 0.0659 y0.0004 0.1322 y0.0004 y0.0004 y0.0004 y0.0004 0.0659 y0.0004 y0.0004 y0.0004 y0.0004 y0.0004 y0.0004 0.1322 0.1322
The GCHF method is the result of choosing the one-electron function as integral transforms c i Ž 1 . s H f i Ž 1, a . f 1 Ž a . d a , i s 1, . . . ,n , Ž 1 . where f i are the generator functions ŽSTF in this work., the f i are the weight functions and a is the generator coordinate. The c i are then used to build the Slater determinant and to calculate the mean value of the total energy. Minimizing the total energy with respect to the f i one can obtain the Griffin–Wheeler–HF ŽGWHF. equations H F Ž a , b . y ´ i S Ž a , b . f i Ž b . d b s 0, i s 1, . . . ,n , Ž 2. where ´ i are the HF eigenvalues and F Ž a , b . and SŽ a , b . are, respectively, the Fock and overlap Kernels Žsee Ref. w1x for more details about these kernels.. The GWHF equations are solved by discretization through a technique called integral discretization ŽID. w12x, which preserves the continuous representation Žintegral character. of the generator coordinate ŽGC. method w13,14x. This technique is implemented through a relabelling of the generator coordinate space, i.e. ln a Vs , A)1 , Ž 3. A where A is a scalling parameter determined numerically. The lowest Ž V min . and highest Ž V max . values
F.E. Jorge, R.F. Martinsr Chemical Physics 233 (1998) 1–7
for the generator coordinate are chosen so as to embrace the adequate integration range for the f i . They are related by
V max s V min q Ž N y 1 . DV ,
Ž 4.
where N corresponds to the number of discretization points and D V is the increment. In fact, the choice of the discretization points determines the exponents of the basis functions, and N defines the STF basis set size. For more details about the GCHF formalism see Refs. w1x and w2x. Here we would like to draw the attention to the similarity between the even-tempered formula and the ID technique used in GCHF method. The definition for the even-tempered exponents is ´ i s ab iy1, i s 1, 2, . . . , M with a and b constants Žalthough different for different symmetries.. Then the increment is D ´ s ´ iq1 y ´ i s ab iy1 Ž b y 1. s ´ i Ž b y 1., i.e. the exponent times a constant. In our case ŽEq. Ž3., a i s expŽ A V i . and D a s a iq1 y a i s expŽ A V i .wexpŽ AD V . y 1x s a i wexpŽ AD V . y 1x, again the exponent times a constant. The ID approach contrasts to the procedure of optimizing orbital exponents, common in Roothaan–Hartree–Fock calculations, i.e. variational discretization ŽVD.. An extensive discussion about these two approaches ŽID and VD. and their applications in two problems having exact solutions Žthe harmonic oscillator and the hydrogen atom. is given in Ref. w12x.
3. Results and discussion The HF SCF calculations are performed, for atoms H Ž Z s 1. through Xe Ž Z s 54., by employing the STF exponents generated with the GCHF method w1x. The universal STF basis set exponents generated in this work can be easily reproduced by using Eqs. Ž3. and Ž4. and Table 1. The starting value of the exponent Ž V min . and the number of basis function exponents Ž N . for s, p and d symmetries are shown in Table 1 for each atom studied here. The optimal increment value Ž D V . calculated here for all symmetries of all atoms ŽH to Xe. is equal to 0.0663. With this value and with the values of V min shown in Table 1, we find a unique set of STF exponents. We note that for B to Xe the STF exponents corresponding to p or p and d symmetries of a given atom
3
are subsets of the STF exponents of the s symmetry. For example, in the case of Co Ž Z s 27., one can see that if we add to V min of the s symmetry Žy0.0667. the value 3D V Ž3 = 0.0663 s 0.1989. we obtain the V min of the p symmetry Ž0.1322., and if we add the value 2D V Ž2 = 0.0663 s 0.1326. to V min of the s symmetry we obtain the V min of the d symmetry Ž0.0659.. From Table 1 we can also observe that the s-symmetry exponents of all atoms ŽH to Xe. are subsets of the set generated by the union of the s-symmetry exponents of the Li and Pd atoms. Thus, we have a unique set ŽUBS. of STF exponents for the atoms from H through Xe. In Table 2 we show the HF total ground state energy results for the atoms H Ž Z s 1. to Xe Ž Z s 54. computed with our UBS. The UBS exponents used to obtain the HF energies shown in Table 2 were generated by employing the ID technique of the GCHF method. As we have seen before, the exponentes can be easily reproduced by using the discretization parameters, V min and D V Ždisplayed in Table 1., and the STF number given in Table 2 for each atom under study Žsee the fourth column.. The optimal scaling parameter A Žsee Eq. Ž3.. found here was 6.0. It is important to remark that the GCHF method is one of the more flexible formalisms in the design of UBS, and that the ID technique of the GCHF method allows us to generate easily UBSs that are able to describe the HF total energy for a large number of atoms with good accuracy w5,6x. Table 2 presents a comparison of the total energy results obtained by different methods: our UBS, the even-tempered basis sets Žan optimized basis set for each atom studied here. w15x, the fully-optimized wave functions Ža fully-optimized basis set for each atom in study. w16,17x and the numerical HF ŽNHF. functions w17,18x. Our total energy values Žsee the fifth colmn of Table 2. agree with those of the NHF Žsee the twelveth column of Table 2. within errors equal to or lower than 6 = 10y6 , 8 = 10y5 , 9 = 10y4 and 4 = 10y3 hartree for H to Ne, Na to Cl, Ar to Rb and Sr to Xe, respectively. For the atoms He Ž Z s 2. to Mg Ž Z s 12., we found energy values which in general are equal to the corresponding values obtained with the reoptimized Clementi–Roetti basis sets Žsee the seventh column of Table 2. w16x. For the atoms from Rb Ž Z s 37. through Xe Ž Z s 54., our energy values are lower than the corresponding ones
4
Z
Atom
Configuration
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
H Ž2 S . He Ž1 S . Li Ž2 S . Be Ž1 S . B Ž2 P . C Ž3 P . N Ž4S . O Ž3 P . F Ž2 P . Ne Ž1 S . Na Ž2 S . Mg Ž1 S . Al Ž2 P . Si Ž3 P . P Ž4S . S Ž3 P . Cl Ž2 P . Ar Ž1 S . K Ž2 S . Ca Ž1 S . Sc Ž2 D . Ti Ž3 F . V Ž4F . Cr Ž7S . Mn Ž6 S .
1sŽ1. 1sŽ2. wHex2sŽ1. wHex2sŽ2. wHex2sŽ2.2pŽ1. wHex2sŽ2.2pŽ2. wHex2sŽ2.2pŽ3. wHex2sŽ2.2pŽ4. wHex2sŽ2.2pŽ5. wHex2sŽ2.2pŽ6. wNex3sŽ1. wNex3sŽ2. wNex3sŽ2.3pŽ1. wNex3sŽ2.3pŽ2. wNex3sŽ2.3pŽ3. wNex3sŽ2.3pŽ4. wNex3sŽ2.3pŽ5. wNex3sŽ2.3pŽ6. wArx4sŽ1. wArx4sŽ2. wArx4sŽ2x3dŽ1. wArx4sŽ2.3dŽ2. wArx4sŽ2.3dŽ3. wArx4sŽ1.3dŽ5. wArx4sŽ2.3dŽ5.
Present UBSa
Koga et al.b
Koga et al.c
No. STF
No. STF Energy
No. STF
Energy
No. STF
Energy
y 5s 6s 6s 6s4p 6s4p 6s4p 6s4p 6s4p 6s4p 8s5p 8s5p 8s8p 8s8p 8s8p 8s8p 8s8p 8s8p 11s6p 11s6p 11s6p5d 11s6p5d 11s6p5d 11s6p5d 11s6p5d
4s 7s 7s 7s5p 7s5p 7s5p 7s5p 7s5p 7s5p 10s5p 10s5p 10s8p 10s8p 10s8p 10s8p 10s8p 10s8p 11s7p 11s7p 11s7p5d 11s7p5d 11s7p5d 11s7p5d 11s7p5d
2.8616799 7.4327259 14.573021 24.529059 37.688615 54.400928 74.809390 99.409336 128.54708 161.85887 199.61459 241.87668 288.85434 340.71875 397.50483 459.48201 526.81745 599.16464 676.75803 759.73555 848.40581 942.88413 1043.3560 1149.8660
y 4s 7s 7s 7s5p 7s5p 7s5p 7s5p 7s5p 7s5p 10s5p 10s5p 10s8p 10s8p 10s8p 10s8p 10s8p 10s8p 11s7p 11s7p 11s7p5d 11s7p5d 11s7p5d 11s7p5d 11s7p5d
y 2.8616 799 95 7.432726928 14.57302316 24.52906072 37.68861894 54.40093418 74.80939842 99.40934931 128.5470980 161.8589113 199.6146361 241.8767072 288.8543625 340.7187808 397.5048957 459.4820721 526.8175125 599.1647847 676.7581825 759.7357145 848.4059932 942.8843337 1043.356371 1149.866247
Energy
7s 0.5 8s 2.861679958 12s 7.432726260 11s 14.57302218 10s6p 24.52905827 12s7p 37.68861575 11s8p 54.40093126 12s8p 74.80939710 12s8p 99.40934391 11s8p 128.5470919 12s8p 161.8588971 12s8p 199.6146245 12s9p 241.8766651 12s9p 288.8542793 12s9p 340.7187358 12s9p 397.5048569 12s9p 459.4819884 12s10p 526.8174074 12s10p 599.1644898 12s10p 676.7577952 13s10p8d 759.7355419 13s9p7d 848.4057978 13s10p7d 942.8841144 12s10p7d 1043.356024 12s10p8d 1149.865930
y 2.8616800 7.4327258 14.573021 24.529058 37.688616 54.400931 74.809395 99.409344 128.54709 161.85891 199.61463 241.87670 288.85436 340.71877 397.50489 459.48207 526.81751 599.16470 676.75810 759.73563 848.40592 942.88426 1043.3563 1149.8662
Koga et al.d
NHF e 0.5 f 2.861679996 7.432726931 14.57302317 24.52906073 37.68861896 54.40093421 74.80939847 99.40934939 128.5470981 161.8589116 199.6146364 241.8767072 288.8543625 340.7187810 397.5048959 459.4820724 526.8175128 599.1647868 676.7581859 759.7357180 848.4059970 942.8843377 1043.356376 1149.866252
F.E. Jorge, R.F. Martinsr Chemical Physics 233 (1998) 1–7
Table 2 Hartree–Fock total ground state energies Žsign reversed. in hartree of the atoms H Ž Zs1. through Xe Ž Zs 54.
a
Fe Ž5D . Co Ž4F . Ni Ž3 F . Cu Ž2 S . Zn Ž1 S . Ga Ž2 P . Ge Ž3 P . As Ž4S . Se Ž3 P . Br Ž2 P . Kr Ž1 S . Rb Ž2 S . Sr Ž1 S . Y Ž2 D . Zr Ž3 F . Nb Ž6 D . Mo Ž7S . Tc Ž6 S . Ru Ž5F . Rh Ž4F . Pd Ž1 S . Ag Ž2 S . Cd Ž1 S . In Ž2 P . Sn Ž3 P . Sb Ž4S . Te Ž3 P . I Ž2 P . Xe Ž1 S .
wArx4sŽ2.3dŽ6. wArx4sŽ2.3dŽ7. wArx4sŽ2.3dŽ8. wArx4sŽ1.3dŽ10. wArx4sŽ2.3dŽ10. wArx4sŽ2.3dŽ10.4pŽ1. wArx4sŽ2.3dŽ10.4pŽ2. wArx4sŽ2.3dŽ10.4pŽ3. wArx4sŽ2.3dŽ10.4pŽ4. wArx4sŽ2.3dŽ10.4pŽ5. wArx3sŽ2.3dŽ10.4pŽ6. wKrx5sŽ1. wKrx5sŽ2. wKrx5sŽ2.4dŽ1. wKrx5sŽ2.4dŽ2. wKrx5sŽ1.4dŽ4. wKrx5sŽ1.4dŽ5. wKrx5sŽ2.4dŽ5. wKrx5sŽ1.4dŽ7. wKrx5sŽ1.4dŽ8. wKrx4dŽ10. wKrx5sŽ1.4dŽ10. wKrx5sŽ2.4dŽ10. wKrx5sŽ2.4dŽ10.5pŽ1. wKrx5sŽ2.4dŽ10.5pŽ2. wKrx5sŽ2.4dŽ10.5pŽ3. wKrx5sŽ2.4dŽ10.5pŽ4. wKrx5sŽ2.4dŽ10.5pŽ5. wKrx5sŽ2.4dŽ10.5pŽ6.
12s10p8d 13s10p7d 13s10p8d 13s10p8d 13s10p8d 13s11p7d 13s11p8d 13s10p8d 13s10p8d 13s10p9d 13s10p9d 13s10p9d 13s9p7d 13s10p9d 14s10p9d 14s10p9d 14s10p9d 14s10p9d 14s10p9d 14s10p9d 14s10p9d 14s10p9d 14s10p9d 13s12p9d 13s11p9d 13s11p9d 13s11p8d 13s11p9d 13s11p9d
1262.443305 1381.414271 1506.870600 1638.963381 1777.847646 1923.260375 2075.359051 2234.237907 2399.866853 2572.440620 2752.054305 2938.356520 3131.544107 3331.682513 3538.993790 3753.596405 3975.548004 4204.787249 4441.537682 4685.879681 4937.918924 5197.695724 5465.130092 5740.164943 6022.927001 6313.482137 6611.780991 6917.978055 7232.135288
HF total energies obtained by using our universal basis set. HF total energies obtained from Ref. w16x. c HF total energies obtained from Ref. w15x. d HF total energies obtained from Ref. w17x. e Numerical HF total energies obtained from Ref. w17x. f Numerical HF total energies obtained from Ref. w18x. b
11s6p5d 11s6p5d 11s6p5d 11s6p5d 11s6p5d 10s9p5d 10s9p5d 10s9p5d 10s9p5d 10s9p5d 10s9p5d 11s7p3d 11s7p3d 11s7p5d 11s7p5d 11s7p5d 11s7p5d 11s7p5d 11s7p5d 11s7p5d 9s7p5d 11s7p5d 11s7p5d 11s9p5d 11s9p5d 11s9p5d 11s9p5d 11s9p5d 11s9p5d
1262.4436 1381.4145 1506.8709 1638.9637 1777.8481 1923.2609 2075.3597 2234.2386 2399.8676 2572.4413 2752.0549 2938.3531 3131.5417 3331.6807 3538.9914 3753.5917 3975.5430 4204.7839 4441.5310 4685.8726 4937.9091 5197.6890 5465.1253 5740.1638 6022.9271 6313.4813 6611.7803 6917.9773 7232.1350
11s7p5d 11s7p5d 11s7p5d 11s7p5d 11s7p5d 11s10p5d 11s10p5d 11s10p5d 11s10p5d 11s10p5d 11s10p5d 13s10p5d 13s10p5d 13s10p8d 13s10p8d 13s10p8d 13s10p8d 13s10p8d 13s10p8d 13s10p8d 13s10p8d 13s10p8d 13s10p8d 13s12p8d 13s12p8d 13s12p8d 13s12p8d 13s12p8d 13s12p8d
1262.4433 1381.4141 1506.8703 1638.9629 1777.8475 1923.2605 2075.3593 2234.2382 2399.8671 2572.4407 2752.0543 2938.3567 3131.5451 3331.6837 3538.9946 3753.5972 3975.5489 4204.7881 4441.5386 4685.8806 4937.9204 5197.6971 5465.1320 5740.1681 6022.9309 6313.4847 6611.7834 6917.9803 7232.1378
11s7p5d 11s7p5d 11s7p5d 11s7p5d 11s7p5d 11s10p5d 11s10p5d 11s10p5d 11s10p5d 11s10p5d 11s10p5d 13s10p5d 13s10p5d 13s10p8d 13s10p8d 13s10p8d 13s10p8d 13s10p8d 13s10p8d 13s10p8d 11s10p8d 13s10p8d 13s10p8d 13s12p8d 13s12p8d 13s12p8d 13s12p8d 13s12p8d 13s12p8d
1262.443660 1381.414547 1506.870902 1638.963729 1777.848108 1923.261005 2075.359729 2234.238650 2399.867608 2572.441329 2752.054974 2938.357450 3131.545683 3331.684164 3538.995058 3753.597720 3975.549492 4204.788729 4441.539479 4685.881695 4937.921014 5197.698464 5465.133133 5740.169148 6022.931687 6313.485312 6611.784050 6917.980887 7232.138355
1262.443665 1381.414553 1506.870908 1638.963742 1777.848116 1923.261010 2075.359734 2234.238654 2399.867612 2572.441333 2752.054977 2938.357454 3131.545686 3331.684170 3538.995065 3753.597728 3975.549500 4204.788737 4441.539488 4685.881704 4937.921024 5197.698473 5465.133143 5740.169156 6022.931695 6313.485321 6611.784059 6917.980896 7232.138364
F.E. Jorge, R.F. Martinsr Chemical Physics 233 (1998) 1–7
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5
6
F.E. Jorge, R.F. Martinsr Chemical Physics 233 (1998) 1–7
obtained by Koga et al. w16x. For the remaining atoms our energy results are larger than those of Koga et al. w16x. Although we have used a larger basis set than the reoptimized one of Clementi and Roetti, this result is surprising, since the reoptimized Clementi–Roetti wave functions are fully-optimized for each atom individually, while we have used a UBS Ža unique set of exponents that describe all atoms studied.. It is important here to note that we are working with universal STF basis set, and thus the size of the basis set is necessarily larger than the fully-optimized ones, in order to embrace all atoms studied. We also point out that the UBS generated by us with the GCHF method required, for example, only three nonlinear parameters for the third- and fourth-row atoms Žsee Eq. Ž4.., except for K and Ca which needed only two Žthe parameter D V is not added because it has the same value Ž0.0663. for all atoms studied., whereas for Xe, for example, the Clementi–Roetti function w19x has 25 nonlinear parameters. For the atoms from He Ž Z s 2. through Mg Ž Z s 12., the HF energy values obtained with our UBS are in general lower than the corresponding values obtained with the even-tempered basis sets Žsee the ninth column of Table 2. w15x, and for Al Ž Z s 13. through Kr Ž Z s 36. our results are, in general, slightly larger. Finally, for Rb Ž Z s 37. through Xe Ž Z s 54. our energy values are worse than those obtained with the even-tempered basis sets w15x. Koga et al. w17x, in addition to optimizing the exponents of each symmetry for each atom, has made a careful choice of the principal quantum numbers n4 , and theirs energies Žsee the eleventh column of Table 2. are approximately equal to the corresponding NHF values for all atoms and better than our results. However, the apparent lack of regularity w20x in the optimal n4 , makes it difficult to extend such an optimized basis set by adding more diffuse or tight basis functions that may be required in the description of properties others than the energy. On the other hand, there is no ambiguity in extending our UBS generated by GCHF method that uses only functions with the lowest value of n; one simply extends the arithmetic sequence of Eq. Ž4.. The importance of the choice of n4 has been demonstrated recently for He–Xe within single-zeta w20x approximation.
In all cases studied here our energy values Žexcept for Ar, Ca, Fe and As. are lower than recent HF energy results of Jorge et al. w5x calculated with a large UGBS. Also, they are more accurate than Žexcept for Ne. the corresponding total energy values obtained by a previous universal STF basis set of Ref. w4x, where the basis set size used for all atoms from H through Xe is equal to 12s10p10d. Thus, for the atoms H Ž Z s 1. through Se Ž Z s 34. our UBS size is slightly smaller than the UBS size of Ref. w4x and for Br Ž Z s 35. through Xe Ž Z s 54. is slightly larger. The main difference between the two UBSs is that in Ref. w4x the authors have used for the same symmetry of all atoms the same value of V min , i.e. V min Žs. s y0.11, V min Žp. s 0.00 and V min Žd. s y0.11, whereas we have searched the best V min for each symmetry of each atom Žsee Table 1.. Besides this, our D V value Ž0.0663. is different from theirs Ž0.07.. Here, we would like to say that the HF energies obtained with our universal STF basis set for the atoms studied are the best obtained so far, at least using a UBS in HF calculations.
4. Concluding remarks In summary, we have reported accurate universal STF basis set for the atoms H through Xe generated with the GCHF method. The greater size of our UBS compared to a fully-optimized basis set is compensated by a marked reduction in the number of nonlinear parameters to be optimized and by its ease to be extended. For all atoms studied here, the HF ground state energies obtained with our UBS are competitive with those obtained by fully-optimized basis sets and NHF calculations. The universal STF basis set generated in this work can also be considered as more accurate than the UBSs previously presented in the literature.
Acknowledgements We would like to acknowledge the financial support by CNPq ŽBrazilian agency..
F.E. Jorge, R.F. Martinsr Chemical Physics 233 (1998) 1–7
References w1x J.R. Mohallem, R.M. Dreizler, M. Trsic, Int. J. Quantum Chem. Symp. 20 Ž1986. 45. w2x J.R. Mohallem, M. Trsic, J. Chem. Phys. 86 Ž1987. 5043. w3x H.F.M. Da Costa, M. Trsic, J.R. Mohallem, Mol. Phys. 62 Ž1987. 91. w4x A.B.F. Da Silva, M. Trsic, Mol. Phys. 68 Ž1989. 433. w5x F.E. Jorge, E.V.R. Castro, A.B.F. da Silva, Chem. Phys. 216 Ž1997. 317. w6x F.E. Jorge, E.V.R. Castro, A.B.F. da Silva, J. Comp. Chem. 18 Ž1997. 1565. w7x O. Matsuoka, S. Huzinaga, Chem. Phys. Lett. 140 Ž1987. 567. w8x D.M. Silver, W.C. Nieuwpoort, Chem. Phys. Lett. 57 Ž1978. 421. w9x D.M. Silver, S. Wilson, W.C. Nieuwpoort, Int. J. Quantum Chem. 14 Ž1978. 635.
w10x w11x w12x w13x w14x w15x w16x w17x w18x w19x w20x
7
D.L. Cooper, S. Wilson, J. Chem. Phys. 76 Ž1982. 6088. D.L. Cooper, S. Wilson, J. Chem. Phys. 78 Ž1983. 2456. J.R. Mohallem, Z. Phys. D 3 Ž1986. 339. J.J. Griffin, J.A. Wheeler, Phys. Rev. 108 Ž1957. 311. D.L. Hill, J.A. Wheeler, Phys. Rev. 89 Ž1953. 1102. T. Koga, H. Tatewaki, A.J. Thakkar, Theor. Chim. Acta 88 Ž1994. 273. T. Koga, H. Tatewaki, A.J. Thakkar, Phys. Rev. A 47 Ž1993. 4510. T. Koga, S. Watanabe, K. Kanayama, R. Yasuda, A.J. Takkar, J. Chem. Phys. 103 Ž1995. 3000. C. Froese–Fisher, The Hartree–Fock Method for Atoms ŽWiley, New York, 1977.. E. Clementi, C. Roetti, At. Data Nuc. Data Tables 14 Ž1974. 177. T. Koga, A.J. Thakkar, Theor. Chim. Acta 85 Ž1993. 363.
Accurate universal basis set for H through Xe for Hartree–Fock calculations F.E. Jorge ) , R.F. Martins Departamento de Fısica — CCE, UniÕersidade Federal do Espırito ´ ´ Santo, 29060-900 Vitoria, ´ ES, Brazil Received 22 July 1997
Abstract We have applied the generator coordinate Hartree–Fock method to generate Slater-type functions for the atoms from H through Xe. The Griffin–Hill–Wheeler–Hartree–Fock equations are integrated numerically generating a universal basis set for these atoms. When compared with the values obtained by Koga et al. wPhys. Rev. A 47 Ž1993. 4510x using a smaller but fully-optimized basis sets, it is generally observed that our Hartree–Fock ground state energies are equal for the atoms from He Ž Z s 2. to Mg Ž Z s 12., larger for the atoms from Al Ž Z s 13. to Kr Ž Z s 36., and lower for the atoms from Rb Ž Z s 37. to Xe Ž Z s 54.. Moreover, our energy results compare satisfactorily with the corresponding atom-optimized and numerical Hartree–Fock calculations of Koga et al. wJ. Chem. Phys. 103 Ž1995. 3000x. q 1998 Elsevier Science B.V. All rights reserved.
1. Introduction In 1986 Mohallen et al. w1x introduced a generator coordinate version of the Hartree–Fock ŽGCHF. method. One of the first applications of the GCHF method was the generation of nonrelativistic Gaussian- and Slater-type functions ŽGTF an STF, respectively. for the atoms of the first and second rows of the Periodic Table w2,3x. Later, da Silva and Trsic presented GTF and STF universal basis sets ŽUBS. encompassing the atoms from H through Xe for atomic and molecular calculations w4x, and recently the GCHF method was tested successfully in the generation of large universal Gaussian basis sets ŽUGBSs. for the atoms H Ž Z s 1. through La Ž Z s 57. w5x and for the heavy atoms from Ce Ž Z s 58. through Lr Ž Z s 103. w6x. We recall that the UGBSs )
Corresponding author. E-mail: [email protected]
w5,6x can be easily used as starting basis sets in relativistic calculations Žsee Ref. w7x.. The first unique set of exponents for various different atoms was found with the even tempered STF basis by Silver et al. w8,9x; applications of universal atomic basis sets to molecular systems were initiated by Cooper and Wilson w10,11x. Another practical application of universal atomic basis sets could be the storage of an unique set of integrals to be used in configuration interaction or perturbation Žinternal or external. calculations for different atoms. Hartree–Fock ŽHF. calculations consider the nuclei represented as particle points. In this case, STF have the correct functional form to describe the nonrelativistic wave function of the atoms near the origin, but they are not particularly suitable for analytic self-consistent field ŽSCF. molecular calculations. On the other hand, GTF are useful in the
0301-0104r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 1 - 0 1 0 4 Ž 9 8 . 0 0 1 3 6 - 0
F.E. Jorge, R.F. Martinsr Chemical Physics 233 (1998) 1–7
2
Table 1 Discretization parametersa of the universal basis set ŽUBS. Z
Atom
N
V minŽs.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51
H He Li Be B C N O F Ne Na Mg Al Si P S Cl Ar K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb
7s 8s 12s 11s 10s6p 12s7p 11s8p 12s8p 12s8p 11s8p 12s8p 12s8p 12s9p 12s9p 12s9p 12s9p 12s9p 12s10p 12s10p 12s10p 13s10p8d 13s9p7d 13s10p7d 12s10p7d 12s10p8d 12s10p8d 13s10p7d 13s10p8d 13s10p8d 13s10p8d 13s11p7d 13s11p8d 13s10p8d 13s10p8d 13s10p9d 13s10p9d 13s10p9d 13s9p7d 13s10p9d 14s10p9d 14s10p9d 14s10p9d 14s10p9d 14s10p9d 14s10p9d 14s10p9d 14s10p9d 14s10p9d 13s12p9d 13s11p9d 13s11p9d
y0.1330 y0.1330 y0.3319 y0.1993 y0.0667 y0.1330 y0.0667 y0.0667 y0.0667 y0.0004 y0.1993 -0.0667 y0.0667 y0.0667 y0.0667 y0.0667 y0.0667 y0.0004 y0.1330 y0.1330 y0.1330 y0.1330 y0.1330 y0.1330 y0.0667 y0.0667 y0.0667 y0.0667 y0.0667 y0.0667 y0.0667 y0.0667 y0.0004 y0.0004 y0.0004 y0.0004 y0.1330 y0.1330 y0.0667 y0.0667 y0.0667 y0.0667 y0.0667 y0.0667 y0.0667 y0.0004 y0.0667 y0.0667 y0.0667 y0.0004 y0.0004
V minŽp.
V minŽd.
Table 1 Žcontinued. Z
Atom
N
V minŽs.
V minŽp.
V minŽd.
52 53 54
Te I Xe
13s11p8d 13s11p9d 13s11p9d
y0.0004 y0.0004 y0.0004
y0.0004 y0.0004 y0.0004
0.1322 0.1322 0.0659
a
y0.0667 y0.0667 y0.0667 y0.0667 y0.0667 y0.0004 0.0659 y0.0004 y0.0667 y0.0667 y0.0004 y0.0004 y0.0667 y0.0004 y0.0004 y0.0004 y0.0004 y0.0004 y0.0667 y0.0004 0.0659 0.0659 0.1322 0.1322 0.1322 0.0659 y0.0667 y0.0667 y0.0004 y0.0004 y0.0004 y0.0004 y0.0004 0.0659 0.0659 0.0659 0.0659 0.0659 0.1322 0.1322 0.1322 0.1322 0.1322 0.1322 y0.0667 y0.0004 y0.0004
The increment value Žsee Eq. Ž4.. used for all symmetries of all atoms ŽH to Xe. is equal to D V s 0.0663.
evaluation of multi-center integrals in molecules but they do not possess the correct functional behavior near the origin. In this work we present a highly accurate universal STF basis set for H to Xe generated with the GCHF method. 2. Computational procedures
y0.0004 y0.0004 y0.0004 y0.0004 0.0659 0.0659 0.0659 y0.0004 y0.0004 0.0659 0.0659 0.1322 0.1322 0.1322 0.1322 0.0659 y0.0004 0.1322 y0.0004 y0.0004 y0.0004 y0.0004 0.0659 y0.0004 y0.0004 y0.0004 y0.0004 y0.0004 y0.0004 0.1322 0.1322
The GCHF method is the result of choosing the one-electron function as integral transforms c i Ž 1 . s H f i Ž 1, a . f 1 Ž a . d a , i s 1, . . . ,n , Ž 1 . where f i are the generator functions ŽSTF in this work., the f i are the weight functions and a is the generator coordinate. The c i are then used to build the Slater determinant and to calculate the mean value of the total energy. Minimizing the total energy with respect to the f i one can obtain the Griffin–Wheeler–HF ŽGWHF. equations H F Ž a , b . y ´ i S Ž a , b . f i Ž b . d b s 0, i s 1, . . . ,n , Ž 2. where ´ i are the HF eigenvalues and F Ž a , b . and SŽ a , b . are, respectively, the Fock and overlap Kernels Žsee Ref. w1x for more details about these kernels.. The GWHF equations are solved by discretization through a technique called integral discretization ŽID. w12x, which preserves the continuous representation Žintegral character. of the generator coordinate ŽGC. method w13,14x. This technique is implemented through a relabelling of the generator coordinate space, i.e. ln a Vs , A)1 , Ž 3. A where A is a scalling parameter determined numerically. The lowest Ž V min . and highest Ž V max . values
F.E. Jorge, R.F. Martinsr Chemical Physics 233 (1998) 1–7
for the generator coordinate are chosen so as to embrace the adequate integration range for the f i . They are related by
V max s V min q Ž N y 1 . DV ,
Ž 4.
where N corresponds to the number of discretization points and D V is the increment. In fact, the choice of the discretization points determines the exponents of the basis functions, and N defines the STF basis set size. For more details about the GCHF formalism see Refs. w1x and w2x. Here we would like to draw the attention to the similarity between the even-tempered formula and the ID technique used in GCHF method. The definition for the even-tempered exponents is ´ i s ab iy1, i s 1, 2, . . . , M with a and b constants Žalthough different for different symmetries.. Then the increment is D ´ s ´ iq1 y ´ i s ab iy1 Ž b y 1. s ´ i Ž b y 1., i.e. the exponent times a constant. In our case ŽEq. Ž3., a i s expŽ A V i . and D a s a iq1 y a i s expŽ A V i .wexpŽ AD V . y 1x s a i wexpŽ AD V . y 1x, again the exponent times a constant. The ID approach contrasts to the procedure of optimizing orbital exponents, common in Roothaan–Hartree–Fock calculations, i.e. variational discretization ŽVD.. An extensive discussion about these two approaches ŽID and VD. and their applications in two problems having exact solutions Žthe harmonic oscillator and the hydrogen atom. is given in Ref. w12x.
3. Results and discussion The HF SCF calculations are performed, for atoms H Ž Z s 1. through Xe Ž Z s 54., by employing the STF exponents generated with the GCHF method w1x. The universal STF basis set exponents generated in this work can be easily reproduced by using Eqs. Ž3. and Ž4. and Table 1. The starting value of the exponent Ž V min . and the number of basis function exponents Ž N . for s, p and d symmetries are shown in Table 1 for each atom studied here. The optimal increment value Ž D V . calculated here for all symmetries of all atoms ŽH to Xe. is equal to 0.0663. With this value and with the values of V min shown in Table 1, we find a unique set of STF exponents. We note that for B to Xe the STF exponents corresponding to p or p and d symmetries of a given atom
3
are subsets of the STF exponents of the s symmetry. For example, in the case of Co Ž Z s 27., one can see that if we add to V min of the s symmetry Žy0.0667. the value 3D V Ž3 = 0.0663 s 0.1989. we obtain the V min of the p symmetry Ž0.1322., and if we add the value 2D V Ž2 = 0.0663 s 0.1326. to V min of the s symmetry we obtain the V min of the d symmetry Ž0.0659.. From Table 1 we can also observe that the s-symmetry exponents of all atoms ŽH to Xe. are subsets of the set generated by the union of the s-symmetry exponents of the Li and Pd atoms. Thus, we have a unique set ŽUBS. of STF exponents for the atoms from H through Xe. In Table 2 we show the HF total ground state energy results for the atoms H Ž Z s 1. to Xe Ž Z s 54. computed with our UBS. The UBS exponents used to obtain the HF energies shown in Table 2 were generated by employing the ID technique of the GCHF method. As we have seen before, the exponentes can be easily reproduced by using the discretization parameters, V min and D V Ždisplayed in Table 1., and the STF number given in Table 2 for each atom under study Žsee the fourth column.. The optimal scaling parameter A Žsee Eq. Ž3.. found here was 6.0. It is important to remark that the GCHF method is one of the more flexible formalisms in the design of UBS, and that the ID technique of the GCHF method allows us to generate easily UBSs that are able to describe the HF total energy for a large number of atoms with good accuracy w5,6x. Table 2 presents a comparison of the total energy results obtained by different methods: our UBS, the even-tempered basis sets Žan optimized basis set for each atom studied here. w15x, the fully-optimized wave functions Ža fully-optimized basis set for each atom in study. w16,17x and the numerical HF ŽNHF. functions w17,18x. Our total energy values Žsee the fifth colmn of Table 2. agree with those of the NHF Žsee the twelveth column of Table 2. within errors equal to or lower than 6 = 10y6 , 8 = 10y5 , 9 = 10y4 and 4 = 10y3 hartree for H to Ne, Na to Cl, Ar to Rb and Sr to Xe, respectively. For the atoms He Ž Z s 2. to Mg Ž Z s 12., we found energy values which in general are equal to the corresponding values obtained with the reoptimized Clementi–Roetti basis sets Žsee the seventh column of Table 2. w16x. For the atoms from Rb Ž Z s 37. through Xe Ž Z s 54., our energy values are lower than the corresponding ones
4
Z
Atom
Configuration
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
H Ž2 S . He Ž1 S . Li Ž2 S . Be Ž1 S . B Ž2 P . C Ž3 P . N Ž4S . O Ž3 P . F Ž2 P . Ne Ž1 S . Na Ž2 S . Mg Ž1 S . Al Ž2 P . Si Ž3 P . P Ž4S . S Ž3 P . Cl Ž2 P . Ar Ž1 S . K Ž2 S . Ca Ž1 S . Sc Ž2 D . Ti Ž3 F . V Ž4F . Cr Ž7S . Mn Ž6 S .
1sŽ1. 1sŽ2. wHex2sŽ1. wHex2sŽ2. wHex2sŽ2.2pŽ1. wHex2sŽ2.2pŽ2. wHex2sŽ2.2pŽ3. wHex2sŽ2.2pŽ4. wHex2sŽ2.2pŽ5. wHex2sŽ2.2pŽ6. wNex3sŽ1. wNex3sŽ2. wNex3sŽ2.3pŽ1. wNex3sŽ2.3pŽ2. wNex3sŽ2.3pŽ3. wNex3sŽ2.3pŽ4. wNex3sŽ2.3pŽ5. wNex3sŽ2.3pŽ6. wArx4sŽ1. wArx4sŽ2. wArx4sŽ2x3dŽ1. wArx4sŽ2.3dŽ2. wArx4sŽ2.3dŽ3. wArx4sŽ1.3dŽ5. wArx4sŽ2.3dŽ5.
Present UBSa
Koga et al.b
Koga et al.c
No. STF
No. STF Energy
No. STF
Energy
No. STF
Energy
y 5s 6s 6s 6s4p 6s4p 6s4p 6s4p 6s4p 6s4p 8s5p 8s5p 8s8p 8s8p 8s8p 8s8p 8s8p 8s8p 11s6p 11s6p 11s6p5d 11s6p5d 11s6p5d 11s6p5d 11s6p5d
4s 7s 7s 7s5p 7s5p 7s5p 7s5p 7s5p 7s5p 10s5p 10s5p 10s8p 10s8p 10s8p 10s8p 10s8p 10s8p 11s7p 11s7p 11s7p5d 11s7p5d 11s7p5d 11s7p5d 11s7p5d
2.8616799 7.4327259 14.573021 24.529059 37.688615 54.400928 74.809390 99.409336 128.54708 161.85887 199.61459 241.87668 288.85434 340.71875 397.50483 459.48201 526.81745 599.16464 676.75803 759.73555 848.40581 942.88413 1043.3560 1149.8660
y 4s 7s 7s 7s5p 7s5p 7s5p 7s5p 7s5p 7s5p 10s5p 10s5p 10s8p 10s8p 10s8p 10s8p 10s8p 10s8p 11s7p 11s7p 11s7p5d 11s7p5d 11s7p5d 11s7p5d 11s7p5d
y 2.8616 799 95 7.432726928 14.57302316 24.52906072 37.68861894 54.40093418 74.80939842 99.40934931 128.5470980 161.8589113 199.6146361 241.8767072 288.8543625 340.7187808 397.5048957 459.4820721 526.8175125 599.1647847 676.7581825 759.7357145 848.4059932 942.8843337 1043.356371 1149.866247
Energy
7s 0.5 8s 2.861679958 12s 7.432726260 11s 14.57302218 10s6p 24.52905827 12s7p 37.68861575 11s8p 54.40093126 12s8p 74.80939710 12s8p 99.40934391 11s8p 128.5470919 12s8p 161.8588971 12s8p 199.6146245 12s9p 241.8766651 12s9p 288.8542793 12s9p 340.7187358 12s9p 397.5048569 12s9p 459.4819884 12s10p 526.8174074 12s10p 599.1644898 12s10p 676.7577952 13s10p8d 759.7355419 13s9p7d 848.4057978 13s10p7d 942.8841144 12s10p7d 1043.356024 12s10p8d 1149.865930
y 2.8616800 7.4327258 14.573021 24.529058 37.688616 54.400931 74.809395 99.409344 128.54709 161.85891 199.61463 241.87670 288.85436 340.71877 397.50489 459.48207 526.81751 599.16470 676.75810 759.73563 848.40592 942.88426 1043.3563 1149.8662
Koga et al.d
NHF e 0.5 f 2.861679996 7.432726931 14.57302317 24.52906073 37.68861896 54.40093421 74.80939847 99.40934939 128.5470981 161.8589116 199.6146364 241.8767072 288.8543625 340.7187810 397.5048959 459.4820724 526.8175128 599.1647868 676.7581859 759.7357180 848.4059970 942.8843377 1043.356376 1149.866252
F.E. Jorge, R.F. Martinsr Chemical Physics 233 (1998) 1–7
Table 2 Hartree–Fock total ground state energies Žsign reversed. in hartree of the atoms H Ž Zs1. through Xe Ž Zs 54.
a
Fe Ž5D . Co Ž4F . Ni Ž3 F . Cu Ž2 S . Zn Ž1 S . Ga Ž2 P . Ge Ž3 P . As Ž4S . Se Ž3 P . Br Ž2 P . Kr Ž1 S . Rb Ž2 S . Sr Ž1 S . Y Ž2 D . Zr Ž3 F . Nb Ž6 D . Mo Ž7S . Tc Ž6 S . Ru Ž5F . Rh Ž4F . Pd Ž1 S . Ag Ž2 S . Cd Ž1 S . In Ž2 P . Sn Ž3 P . Sb Ž4S . Te Ž3 P . I Ž2 P . Xe Ž1 S .
wArx4sŽ2.3dŽ6. wArx4sŽ2.3dŽ7. wArx4sŽ2.3dŽ8. wArx4sŽ1.3dŽ10. wArx4sŽ2.3dŽ10. wArx4sŽ2.3dŽ10.4pŽ1. wArx4sŽ2.3dŽ10.4pŽ2. wArx4sŽ2.3dŽ10.4pŽ3. wArx4sŽ2.3dŽ10.4pŽ4. wArx4sŽ2.3dŽ10.4pŽ5. wArx3sŽ2.3dŽ10.4pŽ6. wKrx5sŽ1. wKrx5sŽ2. wKrx5sŽ2.4dŽ1. wKrx5sŽ2.4dŽ2. wKrx5sŽ1.4dŽ4. wKrx5sŽ1.4dŽ5. wKrx5sŽ2.4dŽ5. wKrx5sŽ1.4dŽ7. wKrx5sŽ1.4dŽ8. wKrx4dŽ10. wKrx5sŽ1.4dŽ10. wKrx5sŽ2.4dŽ10. wKrx5sŽ2.4dŽ10.5pŽ1. wKrx5sŽ2.4dŽ10.5pŽ2. wKrx5sŽ2.4dŽ10.5pŽ3. wKrx5sŽ2.4dŽ10.5pŽ4. wKrx5sŽ2.4dŽ10.5pŽ5. wKrx5sŽ2.4dŽ10.5pŽ6.
12s10p8d 13s10p7d 13s10p8d 13s10p8d 13s10p8d 13s11p7d 13s11p8d 13s10p8d 13s10p8d 13s10p9d 13s10p9d 13s10p9d 13s9p7d 13s10p9d 14s10p9d 14s10p9d 14s10p9d 14s10p9d 14s10p9d 14s10p9d 14s10p9d 14s10p9d 14s10p9d 13s12p9d 13s11p9d 13s11p9d 13s11p8d 13s11p9d 13s11p9d
1262.443305 1381.414271 1506.870600 1638.963381 1777.847646 1923.260375 2075.359051 2234.237907 2399.866853 2572.440620 2752.054305 2938.356520 3131.544107 3331.682513 3538.993790 3753.596405 3975.548004 4204.787249 4441.537682 4685.879681 4937.918924 5197.695724 5465.130092 5740.164943 6022.927001 6313.482137 6611.780991 6917.978055 7232.135288
HF total energies obtained by using our universal basis set. HF total energies obtained from Ref. w16x. c HF total energies obtained from Ref. w15x. d HF total energies obtained from Ref. w17x. e Numerical HF total energies obtained from Ref. w17x. f Numerical HF total energies obtained from Ref. w18x. b
11s6p5d 11s6p5d 11s6p5d 11s6p5d 11s6p5d 10s9p5d 10s9p5d 10s9p5d 10s9p5d 10s9p5d 10s9p5d 11s7p3d 11s7p3d 11s7p5d 11s7p5d 11s7p5d 11s7p5d 11s7p5d 11s7p5d 11s7p5d 9s7p5d 11s7p5d 11s7p5d 11s9p5d 11s9p5d 11s9p5d 11s9p5d 11s9p5d 11s9p5d
1262.4436 1381.4145 1506.8709 1638.9637 1777.8481 1923.2609 2075.3597 2234.2386 2399.8676 2572.4413 2752.0549 2938.3531 3131.5417 3331.6807 3538.9914 3753.5917 3975.5430 4204.7839 4441.5310 4685.8726 4937.9091 5197.6890 5465.1253 5740.1638 6022.9271 6313.4813 6611.7803 6917.9773 7232.1350
11s7p5d 11s7p5d 11s7p5d 11s7p5d 11s7p5d 11s10p5d 11s10p5d 11s10p5d 11s10p5d 11s10p5d 11s10p5d 13s10p5d 13s10p5d 13s10p8d 13s10p8d 13s10p8d 13s10p8d 13s10p8d 13s10p8d 13s10p8d 13s10p8d 13s10p8d 13s10p8d 13s12p8d 13s12p8d 13s12p8d 13s12p8d 13s12p8d 13s12p8d
1262.4433 1381.4141 1506.8703 1638.9629 1777.8475 1923.2605 2075.3593 2234.2382 2399.8671 2572.4407 2752.0543 2938.3567 3131.5451 3331.6837 3538.9946 3753.5972 3975.5489 4204.7881 4441.5386 4685.8806 4937.9204 5197.6971 5465.1320 5740.1681 6022.9309 6313.4847 6611.7834 6917.9803 7232.1378
11s7p5d 11s7p5d 11s7p5d 11s7p5d 11s7p5d 11s10p5d 11s10p5d 11s10p5d 11s10p5d 11s10p5d 11s10p5d 13s10p5d 13s10p5d 13s10p8d 13s10p8d 13s10p8d 13s10p8d 13s10p8d 13s10p8d 13s10p8d 11s10p8d 13s10p8d 13s10p8d 13s12p8d 13s12p8d 13s12p8d 13s12p8d 13s12p8d 13s12p8d
1262.443660 1381.414547 1506.870902 1638.963729 1777.848108 1923.261005 2075.359729 2234.238650 2399.867608 2572.441329 2752.054974 2938.357450 3131.545683 3331.684164 3538.995058 3753.597720 3975.549492 4204.788729 4441.539479 4685.881695 4937.921014 5197.698464 5465.133133 5740.169148 6022.931687 6313.485312 6611.784050 6917.980887 7232.138355
1262.443665 1381.414553 1506.870908 1638.963742 1777.848116 1923.261010 2075.359734 2234.238654 2399.867612 2572.441333 2752.054977 2938.357454 3131.545686 3331.684170 3538.995065 3753.597728 3975.549500 4204.788737 4441.539488 4685.881704 4937.921024 5197.698473 5465.133143 5740.169156 6022.931695 6313.485321 6611.784059 6917.980896 7232.138364
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F.E. Jorge, R.F. Martinsr Chemical Physics 233 (1998) 1–7
obtained by Koga et al. w16x. For the remaining atoms our energy results are larger than those of Koga et al. w16x. Although we have used a larger basis set than the reoptimized one of Clementi and Roetti, this result is surprising, since the reoptimized Clementi–Roetti wave functions are fully-optimized for each atom individually, while we have used a UBS Ža unique set of exponents that describe all atoms studied.. It is important here to note that we are working with universal STF basis set, and thus the size of the basis set is necessarily larger than the fully-optimized ones, in order to embrace all atoms studied. We also point out that the UBS generated by us with the GCHF method required, for example, only three nonlinear parameters for the third- and fourth-row atoms Žsee Eq. Ž4.., except for K and Ca which needed only two Žthe parameter D V is not added because it has the same value Ž0.0663. for all atoms studied., whereas for Xe, for example, the Clementi–Roetti function w19x has 25 nonlinear parameters. For the atoms from He Ž Z s 2. through Mg Ž Z s 12., the HF energy values obtained with our UBS are in general lower than the corresponding values obtained with the even-tempered basis sets Žsee the ninth column of Table 2. w15x, and for Al Ž Z s 13. through Kr Ž Z s 36. our results are, in general, slightly larger. Finally, for Rb Ž Z s 37. through Xe Ž Z s 54. our energy values are worse than those obtained with the even-tempered basis sets w15x. Koga et al. w17x, in addition to optimizing the exponents of each symmetry for each atom, has made a careful choice of the principal quantum numbers n4 , and theirs energies Žsee the eleventh column of Table 2. are approximately equal to the corresponding NHF values for all atoms and better than our results. However, the apparent lack of regularity w20x in the optimal n4 , makes it difficult to extend such an optimized basis set by adding more diffuse or tight basis functions that may be required in the description of properties others than the energy. On the other hand, there is no ambiguity in extending our UBS generated by GCHF method that uses only functions with the lowest value of n; one simply extends the arithmetic sequence of Eq. Ž4.. The importance of the choice of n4 has been demonstrated recently for He–Xe within single-zeta w20x approximation.
In all cases studied here our energy values Žexcept for Ar, Ca, Fe and As. are lower than recent HF energy results of Jorge et al. w5x calculated with a large UGBS. Also, they are more accurate than Žexcept for Ne. the corresponding total energy values obtained by a previous universal STF basis set of Ref. w4x, where the basis set size used for all atoms from H through Xe is equal to 12s10p10d. Thus, for the atoms H Ž Z s 1. through Se Ž Z s 34. our UBS size is slightly smaller than the UBS size of Ref. w4x and for Br Ž Z s 35. through Xe Ž Z s 54. is slightly larger. The main difference between the two UBSs is that in Ref. w4x the authors have used for the same symmetry of all atoms the same value of V min , i.e. V min Žs. s y0.11, V min Žp. s 0.00 and V min Žd. s y0.11, whereas we have searched the best V min for each symmetry of each atom Žsee Table 1.. Besides this, our D V value Ž0.0663. is different from theirs Ž0.07.. Here, we would like to say that the HF energies obtained with our universal STF basis set for the atoms studied are the best obtained so far, at least using a UBS in HF calculations.
4. Concluding remarks In summary, we have reported accurate universal STF basis set for the atoms H through Xe generated with the GCHF method. The greater size of our UBS compared to a fully-optimized basis set is compensated by a marked reduction in the number of nonlinear parameters to be optimized and by its ease to be extended. For all atoms studied here, the HF ground state energies obtained with our UBS are competitive with those obtained by fully-optimized basis sets and NHF calculations. The universal STF basis set generated in this work can also be considered as more accurate than the UBSs previously presented in the literature.
Acknowledgements We would like to acknowledge the financial support by CNPq ŽBrazilian agency..
F.E. Jorge, R.F. Martinsr Chemical Physics 233 (1998) 1–7
References w1x J.R. Mohallem, R.M. Dreizler, M. Trsic, Int. J. Quantum Chem. Symp. 20 Ž1986. 45. w2x J.R. Mohallem, M. Trsic, J. Chem. Phys. 86 Ž1987. 5043. w3x H.F.M. Da Costa, M. Trsic, J.R. Mohallem, Mol. Phys. 62 Ž1987. 91. w4x A.B.F. Da Silva, M. Trsic, Mol. Phys. 68 Ž1989. 433. w5x F.E. Jorge, E.V.R. Castro, A.B.F. da Silva, Chem. Phys. 216 Ž1997. 317. w6x F.E. Jorge, E.V.R. Castro, A.B.F. da Silva, J. Comp. Chem. 18 Ž1997. 1565. w7x O. Matsuoka, S. Huzinaga, Chem. Phys. Lett. 140 Ž1987. 567. w8x D.M. Silver, W.C. Nieuwpoort, Chem. Phys. Lett. 57 Ž1978. 421. w9x D.M. Silver, S. Wilson, W.C. Nieuwpoort, Int. J. Quantum Chem. 14 Ž1978. 635.
w10x w11x w12x w13x w14x w15x w16x w17x w18x w19x w20x
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D.L. Cooper, S. Wilson, J. Chem. Phys. 76 Ž1982. 6088. D.L. Cooper, S. Wilson, J. Chem. Phys. 78 Ž1983. 2456. J.R. Mohallem, Z. Phys. D 3 Ž1986. 339. J.J. Griffin, J.A. Wheeler, Phys. Rev. 108 Ž1957. 311. D.L. Hill, J.A. Wheeler, Phys. Rev. 89 Ž1953. 1102. T. Koga, H. Tatewaki, A.J. Thakkar, Theor. Chim. Acta 88 Ž1994. 273. T. Koga, H. Tatewaki, A.J. Thakkar, Phys. Rev. A 47 Ž1993. 4510. T. Koga, S. Watanabe, K. Kanayama, R. Yasuda, A.J. Takkar, J. Chem. Phys. 103 Ž1995. 3000. C. Froese–Fisher, The Hartree–Fock Method for Atoms ŽWiley, New York, 1977.. E. Clementi, C. Roetti, At. Data Nuc. Data Tables 14 Ž1974. 177. T. Koga, A.J. Thakkar, Theor. Chim. Acta 85 Ž1993. 363.