J. Phys. Chem. 1994, 98, 10098-10101
10098
Choice of Hamiltonian Matrix Elements for SCC-IEH Calculations on Transition-Metal Complexes Jan Blomquist' and Per Kjtsll Department of Physics, Stockholm University, Box 6730, S-133 85 Stockholm, Sweden Received: June 10, 1994@
A model potential is included in the diagonal elements of the Hamiltonian matrix used in IEH calculations on transition-metal complexes. The valence orbital ionization potentials are represented as polynomials of the valence electron population and the atomic number instead of, as is usual, polynomials of the atomic charge. The resulting matrix elements are used in calculations on a number of octahedral transition-metal complexes, and the results are compared for one complex with results obtained from other EH schemes.
0.00
Introduction The extended Huckel (EH) method is based on the idea of an existing effective one-electron operator in some way corresponding to the Fock operator in the SCF methods. However, when using the method, matrix elements are used without explicit reference to any operator. In the original method1,*the diagonal elements are set equal to the valence orbital ionization potentials (denoted VOIP), Ht, = VOIP,, while the off-diagonal elements are equated with some mean value of appropriate diagonal elements. The EH method is modified to an iterative self-consistent charge (SCC) method (denoted IEH) by introducing an explicit charge dependence of the diagonal elements. The charge dependence is normally taken as of first or second order in the atomic charges, see, e.g., Basch et a L 3 s 4 The VOIP of an electron in orbital AO, on atom n with charge q, is then written
1
1;II
E ;;
,
\
-12.00 24 25 Atomic Number
22
20
Figure 1. First parameter ( a l ) in the 3d, 4s, and 4p VOIP fitted to a second-order polynomial in the atomic number 2.
adding a model potential to each metal VOIP resulting in a much improved charge distribution. where superscript c on each constant indicates a particular configuration. By making linear combinations of constants for integer configurations, the VOIP for just any configuration can be calculated. An obvious and intuitive extension of the method of using the atomic charge as a variable in the VOIP is to use the atomic populations. Since all IEH programs contain routines for extracting the charges as well as the populations of the atoms under investigation, e.g., through Mulliken population analysis, the changes needed in the computer code is minimal. While working with this, it was found that the parameters in the "population polynomials" were smooth functions of the atomic number (see below). By making the coefficients in these polynomials functions of the atomic number, the practicability of the polynomials is extended to cover all transition elements from Ti to Ni with the aid of only three polynomials (3d, 4s, and 4p). In transition-metal complexes, the above-described representation for the diagonal elements are normally restricted only to the central metal atom, while the ligand orbitals are described by fixed VOIPs. If, however, even the ligand orbitals are represented by charge dependent VOIPs it is seen (see Table 5) that the charges and populations are less realistic than in the case of fixed ligand VOIPs. This weakness is remedied by
* To whom correspondence should be addressed (NGW@VANA. PHYSTO.SE) . Abstract published in Advance ACS Abstracts, September 1, 1994. @
0022-365419412098-10098$04.50/0
Construction of the Polynomials The 3d, 4s, and 4p ionization potentials were obtained for a large number of configurations by X a calculations (see below). For example, 640 different (3d, 4s, 4p) configurations were used for Fe with the total atomic charge ranging from +3 to 0. These valence orbital ionization potentials were then fitted to secondorder polynomials of the form5
VOIP, = a,
+
+
+ a4n4p+ a5n3: + a6n4? +
~21t3,j 74';
+ 'sn3dn4s + 'gn3dn4p + '10n4sn4p
(2)
where x = 3d, 4s, or 4p and n, are the corresponding populations. Corresponding coefficients for different elements were shown to be smooth functions of the atomic number, see Figure 1. All coefficients were therefore fitted to second-order polynomials of the atomic number
+ ,BF+ r~~
a, = a,
(3)
The resulting coefficients are tabulated in Tables 1-3. An improvement as compared to prior theoretical estimates of atomic ionization energies based on Koopmans' theorem is the use of the Xa "transition state" concept. In the Xa method6s7 a local statistical exchange potential replaces the computationally more cumbersome exchange potential used in Hartree-Fock. 0 1994 American Chemical Society
Hamiltonian Matrix Elements for SCC-IEH Calculations
J. Phys. Chem., Vol. 98, No. 40, 1994 10099
TABLE 1: Parameters in the Z Expansion of the 3d VOIP
7
parameter
a
B
Y
-21.372 8 -1.762 32 -0.734 89 -0.762 99 - 1.50507 -1.284 69 -1.041 22 -1.091 11 -1.726 38 -1.392 16
2.636 890 0.100 138 0.037 827 0.035 158 0.063 459 0.046 514 0.039 563 0.042 286 0.086 060 0.053 527
-0.080 735 0.OOO 754 0.001 103 0.001 255 -0.001 330 -0.001 076 -0.OOO 946 -0.OOO 854 -0.002 061 -0.001 497
TABLE 2: Parameters in the Z Expansion of the 4s VOW uarameter a 5 Y -11.807 80 1.450 680 -0.044 832 0.038 630 0.000 720 -0.735 021 0.013 747 0.000 902 -0.332 811 0.005 712 0.001 131 -0.302 647 0.005 849 -0.392 186 -0.000 283 0.001 477 -0.322 951 -0.000 263 0.000 277 0.183 290 --0.025817 0.020 892 -0.525 792 -0.OOO 462 0.029 727 -0.660 750 -0.000 710 0.017 361 -0.448 163 -0.OOO 549 TABLE 3: Parameters in the Z Expansion of parameter a B -14.441 10 1.662 540 -0.516 395 0.013 510 -0.261 739 0.000 214 -0.103 689 -0.012 976 -0.381 346 0.006 434 -0.540 621 0.018 448 -0.146 146 -0.020 654 -0.526 897 0.021 745 -0.327 649 0.011 982 -0.643 952 0.030 122
the 4p VOIP Y
-0.048 512 0.001 354 0.001 260 0.001 574 -0.OOO 385 -0.OOO 648 0.OOO 143 -0.OOO 539 -0.000 368 -0.000 780
Further, the total energy in the X, method is a function of the occupation and partial derivatives of the total energy with respect to the occupation can be shown to be equal to the energy eigenvalues. By expanding the total energy of two states, differing in the number of electrons by one, in a power series of the occupation it can be sh0wn~9~ that the ionization energy including relaxation effects can be obtained by calculating the energy of a state halfway between the initial and final states. This state is the so called "transition state". Noninteger occupations are interpreted as mixed states due to the statistical nature of the method.
Model Potential The model potential we have used in our calculations to mimic the influence of the ligands has the form of a pointcharge potential
1.00
1 -
0.00
0.00
0.00
2.00
4.00
6.00
Radius R
Figure 2. Broken lines represent the energy of the interactions between the 3d, 4s, and 4p orbitals (on neutral iron) and a charged spherical shell with radius R as a function of R. The solid lines represent the first term in eq 6. All quantities are measured in atomic units. considerably more contracted than the 4s and 4p orbitals. The latter could therefore in a simple picture be said to penetrate the first coordination sphere to a larger extent resulting in a lower interaction between these orbitals and the ligands. This could be modelled by using a larger value for c3d than for ~4~ and ~ 4 An ~ . estimate of the ck values can be obtained if one thinks of the ligands as smeared out into a thin shell-the ligand sphere-centered at the metal atom. The interaction between this shell with charge q L and a valence orbital with charge qk is then
Comparing this expression with eq 4 we see that the ck parameters in this simplified model are given by
Figure 2 illustrates the interaction Wk(R) between a valence electron on neutral Fe and a thin spherical shell with a charge of -1 as a function of the radius R of the shell. The part of the orbital charges laying inside the ligand sphere, Le., the first term in eq 5 , is also illustrated in Figure 2. It can be seen from Figure 2 that all Wk(R)converge toward 1/R as R increases since the last term in eq 5 decreases with R and the first term approaches 1, while for smaller R the interactions are differentiated due to the individual radial dependence of each valence orbital. Although this is an artificial picture it qualitatively supports the use of different ck values and it gives a notion what values these parameters should be given.
Results and Discussion where qi is the charge of ligand i situated a distance from the metal atom. The constants ck, where k denotes a metal orbital, take into account the fact that the valence electrons are not collapsed into the nucleus of the central metal atom but are, to some extent, delocalized and overlap the ligands. In the case of the first-row transition metals we have three different valence orbitals, 3d, 4s, and 4p. Since the radial extension increases with the main quantum number the 3d orbital is
To test our model, we have used the following expression for the diagonal matrix elements for the transition-metal atom:
while for the diagonal elements belonging to the ligands we utilised eq 1 with the constants A , B, and C taken from ref 3.
10100 J. Phys. Chem., Vol. 98, No. 40, 1994
Blomquist and Kjall
TABLE 4: Results for Some Octahedral Complexe@ TiFn3dMe-F n3d n4s
ndP
qMe qL ceg
ne# nt,,
A
VFn3-
3.67 3.72 3.15(3.77) 1.95(2.81) 0.19(0.00) 0.12(0.01) 0.16(0.20) 0.27(0.07) 1.59(1.12) 1.57(1.02) -0.77(-0.69) -0.76(-0.67) -5.19(-9.1) -6.16(-10.5) -7.51(-11.2) -8.16(-12.5) NO) O(0) 1(1) 2(2) 2.32(2.17) 2.00(1.97)
VFn4-
4.06 3.30(3.88) O.OS(0.05)
O.OS(0.14) 1.54(0.93) -0.92(-0.82) -4.35(-11.5) -5.82(-13.0)
CrF& 3.65 4.21(4.76) 0.15(0.06) 0.19(0.21) 1.45(0.97) -0.74(-0.66) -9.57(-11.7) -11.67(-13.7)
O(0)
O(0)
3(3) 1.47(1.49)
3(3) 2.10(1.88)
MnF.5-
Man4-
3.29 4.01 5.27(5.65) 5.41(5.89) 0.17(0.06) 0.12(0.10) 0.24(0.29) 0.1l(0.16) 1.32(1.00) 1.36(0.85) -0.55(-0.33) -0.89(-0.81) -1 1.45(-11.0) -7.84(-13.9) -13.83(-13.8) -S.99(-14.9) 0) 2(2) 3(3) 3(3) 2.38(2.70) 1.15(1.04)
FeFn3-
CoFn3-
NiFn4-
3.63 3.57 6.37(6.76) 7.51(7.72) 0.1S(0.14) 0.16(0.17) 0.20(0.24) O.ll(O.26) 1.25(0.87) 1.22(0.84) -0.71(-0.65) -0.70(-0.64) -12.07(- 13.9) -9.02(-14.7) -13.67(-15.6) -10.12(-16.4) 2(2) 2(2) 3(3) 4(4) 1.60(1.74) l.lO(1.62)
3.78 8.57(8.79) 0.15(0.20) 0.07(0.23) 1.20(0.78) -0.87(-0.80) -9.08(-16.3) -10.03(-17.2) 2(2) 66) 0.95(0.91)
a Data within brackets as well as bond distances are taken from ref 3. dMe-F is the metal-ligand distance. n3dr ndsrand n4, are the valence populations. qMe and q L are the metal and ligand charges. eeg,etlg,negand nlzgare the energies and populations of the eg and tzg molecular orbitals - cegl. Distances, charges, and populations are given in atomic units, while energies are given in electronvolts. and A =
TABLE 5: ComDarison between Different EH Models ADDlied to FeF%- with ~ F . - F = 3.63 a d 6.63 0.21 0.26 0.90 -0.65 -15.3 -17.3 -18.0 -13.4 -8.6 -18.0 -13.4 -8.6 noniterative EH -13.9 -15.6 -18.0 -13.4 -8.6 -18.1 -13.4 -8.6 BVG 6.69 0.21 0.26 0.84 -0.64 -17.4 -15.4 -18.2 -13.6 BK 6.72 0.21 0.26 0.80 -0.63 -8.4 -18.1 -13.5 -8.6 -13.7 -15.5 -9.1 -8.0 -4.0 -16.4 -12.4 -7.7 BVG + Qlig 7.37 0.22 0.26 0.16 -0.53 -13.4 -15.3 -9.1 -7.9 -3.6 -16.5 -12.5 -7.7 BK + Qlig 7.38 0.22 0.25 0.15 -0.53 -11.1 -12.6 -25.0 -17.6 -12.2 -13.1 -10.0 -6.1 BVG+Qlkg Vmol 6.31 0.17 0.18 1.34 -0.72 -12.1 -13.7 -24.8 -17.7 -12.0 -14.1 -10.8 -6.6 BK+ ellg+ Vmol 6.37 0.18 0.20 1.25 -0.71 Populations and charges are expressed in atomic units while all energies are in eV. The following acronymes are used in the table: BVG: Charge-dependent metal VOIPs and fixed ligand VOIPs both from ref 3. No model potential. BK: Population-dependentmetal VOIPs according to eqs 2 and 3. Fixed-ligand VOIPs taken from ref 3. No model potential. Qhg: Charge-dependent ligand VOIPs with coefficients taken from ref 4. V,,,,,,: Model potential according to eq 4.
+
All off-diagonal elements were calculated according to the Wolfsberg-Helmholz recipe:
Hv = -kS,(Hii
+ Hjj)/2
where k = 1.75 has been used throughout the study and Si, is the overlap matrix element between orbitals i and j . Other computational details are given in the Appendix. These modified matrix elements have been utilised in a study of some simple octahedral and tetrahedral transition metal complexes. Table 4 shows some of the results of the octahedral complexes in this study together with results obtained from standard IEH calculation^.^ The geometries are assumed to be perfectly octahedral and the distances are taken from ref 3. For all complexes the set {cgd, cdS, c+} = (0.6,0.4,0.3) has been used so as to make trends more easy to study. In Table 5 we compare different EH schemes for the complex FeFa-3. It is known that standard IEH calculations on transition-metal complexes yield a charge distribution that is too smoothed out. This is clearly illustrated by row BVG and BK in Table 5. The situation is however not improved by also using chargedependent ligand VOIPs only. On the contrary, row 4 and 5 show that, for this complex, the use of charge-dependent metal and ligand VOIPs without a model potential yields a charge distribution that is unrealistic. As is shown by the last two rows in Table 5, the addition of a molecular potential improves things very much. By comparing the charges in Table 4 and 5, we see that this modification gives rise to a much stronger polarization. This degree of polarization is confirmed by comparing these calculated charge distributions by "experimental" populations obtained by combining Mossbauer and XPS data.8-10 What is also clear is that, at least for this complex, there is no major difference between charge and population dependent VOIPs. However, an improvement over charge-
TABLE 6: Orbital Exponents and Coefficients Used in the CalculatioW double-5 single-5 element 51 52 c1 c2 5S.D 4.55 1.80 Ti(+3) 0.462 0.691 1.20 0.498 0.655 1.25 4.75 1.90 V(+3) 0.476 0.706 1.25 4.75 1.70 V(+2) 0.525 4.95 2.00 Cr(+3) 0.629 1.30 Mn(+4) 0.562 1.35 5.15 2.30 0.565 Mn(S2) 0.649 1.35 0.532 5.15 1.90 5.35 2.20 Fe(+3) 0.586 1.40 0.566 co(+3) 0.565 1.45 0.582 5.55 2.30 Ni(+2) 1S O 5.75 2.20 0.589 0.582 2.557* F(-1) All parameters are taken from reference 12 except which is taken from ref 13.
*
dependent VOIPs might result if the iterative scheme is made self-consistent in the population.
Acknowledgment. The authors wish to thank Professor Emeritus Inga Fischer-Hjalmars for many valuable discussions. Appendix: Computational Details All EH calculations were performed with the program QCPE 380." Single STOs were used for the ligand and transitionmetal s and p orbitals while a double-g basis was used for the transition metal 3d orbitals. Exponents and coefficients used are listed in Table 6. VOIP parameters (A, B, and C in eq 1) for the ligand F are taken from ref 4 and are in electronvolts equal to A, = -3.48, B, = -25.51, C, = -40.13, A, = -3.46, B, = -20.52, and C, = -18.65. Iterations were continued until self-consistency in the charge were obtained with a convergence criterion of 0.001. 1% of the previous diagonal element is replaced in each iteration.
Hamiltonian Matrix Elements for SCC-IEH Calculations
References and Notes (1) (2) (3) (4) 3
Hoffman, R. J. Chem. Phys. 1963,39, 1397. Hoffman, R. J. Chem. Phys. 1963,40, 2474. Basch, H.; Viste, A.; Gray, H. B. J. Chem. Phys. 1963,44, 10. Basch, H.; Viste, A.; Gray, H. B. Theor. Chim. Acta (Berlin) 1965,
ACO
J, VJO.
(5) Blomquist, J.; K j a , P. University of Stockholm Internal Report 93-06. (6) Slater, J. C. Phys. Rev. 1951, 81, 385. (7) Slater, J. C. Phys. Rev. 1951, 82, 533. (8) Johansson, L. Y.;Larsson, R.; Blomquist, J.; Cederstrm, C.;
J. Phys. Chem., Vol. 98, No. 40, 1994 10101 Grapengiesser, S.; Helgesson, U.; Moberg, L.; Sundbom, M. Chem. Phys. Lett. 1974,24, 508. (9) Blomquist, J.; Helgesson, U.; Folkesson, B.; Larsson, R. Chem. Phys. 1983, 76, 71. (10) Blomquist, J.; KjXll, P.; Folkesson, B.; Larsson, R. Struct. Chem. 1991,2, 223. (1 1) Quantum Chemistry Program Exchange, Indiana University, Chemistry Department, QCPE Program No. 380. (12) Richardson, J. W.; Nieuwpoort, W. C.; Powell, R. R.; Edgell, W. F. J. Chem. Phys. 1962,36, 1057. (13) Clementi, E.; Raimondi, D. L. J. Chem. Phys. 1963,38, 2686.
