4588
J . Phys. Chem. 1988, 92,4588-4590 of the spectroscopic constants obtained are consistent with this assignment. Additional heavy-metal hydride as well as other systems might be studied by this technique. Many of these systems are of interest for testing relativistic molecular quantum mechanical calculation methods.
exterior to the nozzle. His observation was that this electrode was corroded by the discharge. This technique could perhaps be employed for generating tungsten clusters. In general, the high electronic temperature provided by this arc discharge source presents interesting possibilities for the generation and detection of novel molecular species.
Acknowledgment. We express our thanks to Professor Richard N. Zare for making available his computer code which was employed in the analysis of the data. This research was supported in part by a grant (No. AFOSF-82-0341) from the US Air Force Office of Scientific Research. Registry No. WH, 15478-70-3; D2,7782-39-0; WD, 115075-87-1.
Conclusions We have assigned some observed low-resolution emssion spectra from a plasma obtained from an arc discharge through hydrogen or deuterium, with tungsten electrodes, as due to W H and WD, respectively. All the observed isotope effects and the magnitudes
Extended Slater Basis Sets for the Transltion Metals Scandium through Zinc Lillian M. Hansen and Dennis S. Marynick* Department of Chemistry. University of Texas at Arlington, Arlington, Texas 76019 (Received: October 9, 1987)
Extended Slater basis sets were developed for the transition metals scandium through zinc. These basis sets were generated primarily for use with ab initio methods that generate six Cartesian Gaussians for the d functions. The s-type function generated from the symmetric combination of the x2,y2, and z2 Cartesian Gaussians is used to describe the 3s region, thereby reducing the overall number of functions for molecular calculations.
Introduction Applications of a b initio molecular calculations to transitionmetal complexes are complicated by the presence of d orbitals. The choice of basis functions that accurately describe the d orbitals without demanding an unreasonable amount of computational time presents a dilemma. Generally, Gaussian-type functions are preferred over Slater-type orbitals (STOs), since the two-electron integrals are considerably easier to evaluate over Gaussians. In some implementations of two-electron-integra1 evaluation routines (e.g., the program GAMESS~) six linearly dependent Cartesian d orbitals are retained in the basis set. The sixth component describes an s function, which increases the flexibility of the basis set in the space spanned by the s functions but can also result in a basis set that is redundant. In this paper, we describe Gaussian basis sets for first-row transition metals that directly utilize this sixth d component to describe the 3s region of the metal, thus reducing the number of primitive Gaussians required and eliminating potential problems with linear dependency among the s basis functions. Gaussian basis sets can be arrived at by direct atomic S C F optimization of Gaussian exponents2or by atomic S C F optimization of a Slater basis set, which is then expanded in terms of a limited number of G a u ~ s i a n s . The ~ former procedure suffers from a number of drawbacks. For instance, choosing a proper contraction scheme is sometimes difficult. In addition, energyoptimized Gaussian exponents tend to weight the core regions too heavily and thus may require a large number of primitives for a reasonable description of the valence region. Therefore, we have adopted the latter procedure. We present here a variety of optimized Slater orbital basis sets for first-row transition metals. In all cases, the 3s and 3d exponents (1) Dupuis, M.; Spangler, D.; Wendoloski, J. J. General Atomic and Molecular Electronic Structure System, National Resources for Computations in Chemistry, as modified by: Schmidt, M. W. North Dakota State University; Elbert, T. S . Iowa State University. (2) Poirier, R.; Kari, R.; Csizmadia, I. G . Physical Sciences Data 24, Handbook of Gaussian Basis Sets; Elsevier: New York, 1985. Dobbs, K. D.; Hehre, W. J. J . Comput. Chem. 1987.8, 861. (3) Stewart, R. F. J . Chem. Phys. 1969, 50, 2485.
0022-3654/88/2092-4588$01.50/0
were constrained to be equal. This allows the 3s orbitals to be omitted in the actual molecular calculations, since the 3s region can be described by the sixth component of the Cartesian d functions. The resultant basis sets provide a balanced description of the core and valence regions, while minimizing the number of Gaussian primitives. Eight different basis sets are presented. Each basis set is double-[ in the Is, 2s, 2p, and 4s orbitals. Both doubleand triple-{ d orbital basis sets are described, with a double- or single-{representation of the 3p orbitals. These basis sets were (partially) optimized for the ground state and the 4 ~ ~ 3 d con"+~ figurations of the metals. Finally, a limited discussion of the application of these basis sets to molecular calculations is presented.
Method Extended atomic basis sets were optimized for scandium through zinc by using an atomic S C F program4 based on standard computational procedure^.^ Basis set A is triple-{in the 3s/3d orbitals and double-{everywhere else. The Is, 2s, and 2p exponents were not optimized but were taken directly from the work of Clementi and Roetti.6 Basis set A' is identical with A except that the double-{ representation of the 3p orbitals is replaced by a single-{ 3p exponent optimized for that basis. When supplemented by suitable 4s/4p valence functions, A and A' are particularly useful basis sets for molecular calculations. Basis set B is double-{ in the 3d orbitals, with equivalent double-{ 3s functions. An additional 3s exponent was introduced in this basis set at the atomic level optimizations, primarily to prevent prejudice of the 3s/3d exponents toward description of the 3s orbital. It is quite possible that, for actual molecular calculations, this additional 3s function can be omitted, and the 3s region can be described by the 3s/3d orbitals; however, we have not explicitly tested the omission of the 3s exponent in molecular calculations. (4) Laws, E. (Harvard University, Cambridge, MA) and Walpole Computer Programmers (Massachusetts Department of Corrections, Walple, MA) Atomic Self-Consistent Field Program. September, 1971. ( 5 ) Roothan, C. C.; Bagus, P. S. Methods Comput. Phys. 1963, 2. ( 6 ) Clementi, E.; Roetti, C. At. Data Nucl. Data Tables 1974, 1 4 , 3, 4 .
0 1988 American Chemical Society
The Journal of Physical Chemistry, Vol. 92, No. 16. 1988 4589
Extended Slater Basis Sets for Sc-Zn TABLE I: Extended Basis Sets for Sc through Zn
scandium basis set A (A')
orb. exo
2D4s23d'
titanium 'F 4s23d2
3s/3d
7.792 3.550 1.588 1.464 0.902 3.765 (3.138) 2.370 -759.73220 (-759.647 14) 4.213 1.743 3.130' 1.422 0.884 3.770 (3.138) 2.373 -759.726 65 (-759.641 21)
8.041 3.770 1.752 1.493 0.922 4.044 (3.374) 2.535 -848.401 89 (-848.30246) 4.661 1.983 3.507" 1.521 0.925 4.053 (3.373) 2.541 -848.388 86 (-848.288 73)
4s
3p
tot energy, au
B (B')
3s/3d 4s
3p
tot energy, au
chromium 5D4s23d4 7S 4s13d5
vanadium
4F 4s23d3
8.317 8.588 3.988 4.203 1.88 1 1.997 1.548 1.608 0.948 0.977 4.323 (3.605) 4.600 (3.831) 2.697 2.858 -942.879 54 -1043.30422 (-942.763 58) (-1043.169 83) 5.040 5.400 2.168 2.336 3.781" 4.029" 1.586 1.661 0.952 0.984 4.333 (3.602) 4.611 (3.828) 2.864 2.704 -942.857 33 -1043.27089 (-942.74027) (-1043.13492)
8.260 4.01 1 1.748 1.356 0.867 4.585 (3.796) 2.773 -1043.345 47 (-1043.186 24) 5.102 2.062 3.653" 1.638 0.947 4.592 (3.791) 2.772 -1043.29022 (-1043.12742)
man gan ese
iron
6S 4s23d5 8.822 4.403 2.1 11 1.659 1.ooo 4.880 (4.053) 3.019 -1 149.86000 (-1 149.70488) 5.756 2.505 4.228" 1.702 1.003 4.889 (4.049) 3.022 -1 149.81407 (-1 149.656 85)
5D4s23d6 9.069 4.590 2.174 1.695 1.021 5.155 (4.280) 3.180 -1262.435 82 (-1262.26089) 6.059 2.614 4.445" 1.782 1.040 5.160 (4.275) 3.178 -1262.371 63 (-1262.193 74)
atom basis set A (A')
4s
3p
tot energy, au
B (B')
cobalt 4F 4s23d7
nickel 'F 4s23d8
2D4s23d9
9.329 4.777 2.253 1.740 1.045 5.432 (4.503) 3.341 -1381.40526 (-1381.208 13) 6.375 2.740 4.634" 1.868 1.072 5.443 (4.497) 3.342 -1381.32062 (-1381.11973)
9.587 4.958 2.331 1.778 1.064 5.706 (4.724) 3.499 -1506.86003 (-1506.639 10) 6.693 2.868 4.849" 1.924 1.099 5.722 (4.717) 3.503 -1506.751 95 (-1506.52630)
9.908 5.157 2.417 1.852 1.100 5.981 (4.943) 3.657 -1638.937 51 (-1638.691 07) 7.015 2.998 5.019' 2.029 1.142 5.979 (4.936) 3.645 -1638.80271 (-1638.55062)
orb. exp 3s/3d
3s/3d 4s
3p
tot energy, au
copper 2S 4s13d10
9.651 4.961 2.180 1.596 0.958 5.959 (4.913) 3.573 -1638.94246 (-1638.66498) 6.785 2.760 4.712" 1.875 1.021 5.968 (4.904) 3.565 -1638.74969 (-1638.46220)
zinc
scandium
IS 4s23dI0 10.151 5.322 2.496 1.874 1.110 6.258 (5.161) 3.815 -1777.83400 (-1777.56027) 7.338 3.134 5.184" 2.030 1.145 6.269 (5.152) 3.814 -1777.670 10 (-1777.38952)
4F 4s03d3
titanium 5D4s03d4
6.763 3.150 1.117
6.248 3.059 1.190
3.745 (3.094) 2.255 -759.551 88 (-759.43888) 3.602 1.174 1.469"
4.018 (3.330) 2.410 -848.23261 (-848.101 39) 4.000 1.400 2.039"
3.741 (3.098) 2.255 -759.52937 (-759.41768)
4.026 (3.329) 2.407 -848.18865 (-848.05483)
atom basis set A (A')
orb. exp 3s/3d 3P
tot energy, au
B (B')
3s/3d
3P
tot energy, au
vanadium
chromium
manganese
6S 4s03d5
5D4s03d6
4F 4s03d7
iron SF4s03d8
6.224 3.067 1.250 4.298 (3.558) 2.566 -942.74642 (-942.593 92) 4.438 1.622 3.100' 4.301 2.554 -942.68230 (-942.52360)
10.599 4.135 1.562 4.566 (3.786) 2.713 -1043.061 18 (-1042.885 83) 4.791 1.766 3.410" 4.570 (3.783) 2.707 -1042.98465 (-1042.805 22)
6.814 3.336 1.347 4.845 (4.018) 2.888 -1149.504 10 (-1 149.31205) 5.154 1.920 3.585" 4.854 (4.009) 2.871 -1149.38206 (-1 149.17821)
8.102 4.05 1 1.626 5.113 (4.239) 3.034 -1262.137 10 (-1261.918 61) 5.503 2.068 3.868" 5.125 (4.232) 3.023 -1261.983 55 (-1261.75452)
cobalt
nickel
2D 4s03d9
IS 4s03dI0 8.818 4.457 1.808 5.659 (4.680) 3.345 -1506.63078 (-1506.358 96) 6.191 2.359 4.353" 5.662 (4.671) 3.322 -1506.39581 (-1506.109 51)
8.714 4.367 1.753 5.399 (4.460) 3.200 -1381.11929 (-1380.87468) 5.854 2.217 4.251" 5.382 (4.452) 3.163 -1381.92689 (-1380.67009)
" Represents explicit 3s function. Basis set B' is equivalent to B, with a minimum basis set description of the 3p orbital. For each metal the atomic basis sets were optimized for the ground-state configuration (4s23dnor 4s13dn+l) and for the excited-state configuration 4 ~ ~ 3 d "which + ~ , has been previously shown to yield more accurate state ~plittings.~
Results All of the basis sets are listed in Table I. The values of the total S C F energies are also included. Table I1 presents the values (7) Rappe, A. K.; Srnedely, T. A,; Goddard, W. A. J. Phys. Chem. 1981, 85, 2607.
for the state splittings between the states arising from the 4s2 and 4s0 configurations and compares them to the numerical Hartree-Fock values. As found previ~usly,~ basis sets optimized for the 4s0 configuration produce state splittings much closer to the Hartree-Fock limit. The basis sets optimized for the ground-state configuration demonstrate reasonable trends in the orbital exponents as a function of the atomic number. With the exception of Cr and Mn, the 4s0 basis sets demonstrate similar trends. Although the trends found for the 3d exponents of V through Fe in the Adsobasis sets are not obvious, a thorough search for other local minima always resulted in covergence to the exponents presented in Table I.
4590 The Journal of Physical Chemistry, Vol. 92, No. 16, 1988 TABLE II: State Splittin& basis set optimized for 4sz A A' B B' optimized for 4s0 A A' B B' Hartree-Fock value, eV
Hansen and Marynick
scandium
titanium
vanadium
chromium
manganese
iron
cobalt
nickel
7.43 9.3 1 8.60 10.86
7.54 9.55 9.50 112.1 1
6.59 8.71 9.14 12.00
9.85 12.00 13.50 16.52
13.93 16.08 18.75 21.96
12.08 14.29 17.50 20.79
11.76 14.01 17.96 21.36
10.18 12.52 17.12 20.65
4.35 5.06 4.56 5.28 4.47
3.92 4.79 3.95 4.90 4.25
2.91 3.86 2.80 3.92 3.27
4.84 6.01 4.95 6.26 5.75
8.63 9.66 8.00 9.47 9.15
6.53 7.78 6.59 8.19 7.46
6.01 7.40 6.39 8.16 7.05
4.44 5.96 4.98 6.91 5.47
AE(eV) = 4s23dn- 4s03d"+*.
Atomic S C F calculations in which the triple-{ 3s exponents in Table I were replaced with the optimized double-{ or single-{ 3s exponents taken from Clementi and Roetti6 suggest that our basis sets are of approximate double-{quality in the 3s region. Explicity, for titanium, basis set Ahz produces a total energy of -848.401 90 au and an eigenvalue of -2.8696 au for the 3s orbital. Substitution of optimized double-{ or single-{ 3s exponents results in total energies of -848.40208 and -848.393 51 au, respectively, and eigenvalues of -2.8693 and -2.8672 au for the 3s orbital. These results are typical across the series and indicate that the 3s = 3d constraint does not adversely affect representation of the 3s region. The application of these basis sets to molecular calculations is still in the preliminary stages. We will, therefore, restrict our comments to only a few points. Clearly, molecular calculations will require the selection of the inner-shell representation (singleor double-{), and the determination of the Gaussian expansion lengths for each orbital, the addition of 4p functions in the valence region, and a suitable scaling of the 4s/4p exponents, which are certainly too diffuse for a description of the molecular environment. In our own we generally limit the Is, 2s, and 2p orbitals to a minimum basis set STO 3-G10 description, with exponents taken from Clementi and RoettL6 The accuracy of this procedure has been tested explicitly9 for the reaction Mn(CO)(H) Mn(C0H)
-
Here, the minimum basis set 3G expansion of the inner-shell orbitals on the manganese results in a change in the AE of only -2 kcal/mol (10%) relative to a full double-{ treatment. For orbitals which are a part of double-{ or triple-{ descriptions, we generally employ 2G expansions, although 1G expansions, particularly for the 4s/4p region and 1G expansions of one of the 3d functions, are probably a d e q ~ a t e .We ~ usually test the suit(8) Axe, F. U.; Marynick, D. s. Organometallics 1987, 6, 512. (9) Axe, F. U.;Marynick, D. S. Chem. Phys. Lett. 1987, 141, 455. (10) Hehre, W. J.; Stewart, R. F.; Pople, J. A. J . Chem. Phys. 1969, 51, 2657.
ability of 1G expansions explicitly. The question of the proper choice for valence 4s and 4p exponents is somewhat more difficult and is still being treated on a case by case basis. Starting with the atomic values, we have added 4p functions with exponents equal to the corresponding 4s orbitals and have optimized the scaling factors for the 4p orbitals in a representative group of molecules (TiC14, VF5, Cr(C0)6, HMn(CO),, MnO;, Fe(CO),, Fe(C5H5)*,and Ni(C0)4). This procedure yields a reasonably consistent factor of 1.67 f 0.13, which can be applied to the 4p exponents in Table I. The 4s orbitals are less consistent, and energy optimization sometimes leads to an s orbital basis which is redundant due to excessive overlap between the outermost 3s and innermost 4s orbitals. Further work must be done in this area before "optimum" basis sets in the valence region can be derived. While the basis sets optimized for the 4s0 configurations produce much better state splittings for the atoms, they uniformly yield higher molecular energies. For instance, calculations on Ni(C0)4 using the A4S2and A4s0basis sets with optimized double-{ 4s/4p exponents result in total energies of -1945.3155 and -1944.9756 au, respectively. These results are typical.
Conclusion We have presented Gaussian basis sets derived from leastsquares expansions of STO's which effectively utilized all six Cartesian Gaussian d functions to describe the 3d and 3s regions of first-row transition metals. This procedure yields basis sets which are well-balanced between the core and valence regions and which have a minimum number of primitive functions. Such basis sets should be very useful in conjunction with ab initio programs that retain all six Cartesian d functions. Acknowledgment. We thank the Robert A. Welch Foundation (Grant Y-743) and the Organized Research Fund of the University of Texas at Arlington for their support of this work. Registry No. Sc, 7440-20-2; Ti, 7440-32-6; V, 7440-62-2; Cr, 744047-3; Mn, 7439-96-5; Fe, 7439-89-6; Co, 7440-48-4; Ni, 7440-02-0; Cu, 7440-50-8; Zn, 7440-66-6.
