194
J. Phys. Chem. 1989, 93, 194-200
Variational Theory for g Tensor in Electron Paramagnetic Resonance Experiments: Application to Ferricytochrome c and Aridomyoglobin J. N. Roy, K. C. Mishra, Santosh K. Mishra, and T. P. Das* Department of Physics, State University of New York at Albany, Albany, New York 12222 (Received: February 22, 1988)
A variational procedure employing molecular electronic wave functions from LCAOMO investigations has been utilized to obtain the principal components of the g tensor and orientations of the principal axes in electron paramagnetic resonance (EPR) experiments. Applications have been made to two low-spin heme systems, ferricytochrome c and azidomyoglobin. The variational results provide significant improvement in the agreement with experiment as compared to those from earlier work using perturbation theory. The values for the principal components of the g tensor obtained in this work for ferricytochrome c are (1.731,2.470, 3.172) as compared to (1.250,2.250,3.060) from single-crystalEPR measurements. For azidomyoglobin, the theoretical and experimental values of the principal components are (1.843, 2.528, 2.683) and (1.720, 2.220, 2.800), respectively. Very good agreement is found for the orientationsof the principal axes in azidomyoglobin and the axis corresponding to the largest g component in ferricytochrome c. Possible sources for bridging the remaining gap between theoretical and experimental results are discussed.
I. Introduction The g tensors in heme systems provide very useful information' regarding the nature of bonding between the transition-metal ion and its neighboring ligands in biomolecules. In quantitative terms, the nature of this bonding is reflected in the energy separations of the d-like orbitals and the mixing in the molecular wave functions of the d-orbitals among themselves and with the surrounding ligands. There have indeed been a number of made in tne literature to obtain information regarding these features usLngg-tensor data. Thus, in crystal field approaches, the d-type molecular orbitals are approximated as purely 3d-orbitals. With this approximation, one uses perturbation theory',2 for studying the influence of spin-orbit and orbit-magnetic field levels in spin interactions to determine the splitting of M, = systems in terms of the cubic, axial, and rhombic components of the crystal field, which are kept as disposable parameters. One then determines's2 these parameters by fitting the level splitting to explain observed g-tensor data, thus obtaining useful semiquantitative information regarding the nature of the crystal fields in different systems. The assumption of localized 3d-orbitals on iron is, however, inadequate to explain the observed hyperfine interactions of ligand I4N nuclei as well as protons in the protoporphyrin ring, which have been studied by ENDOR4 and N M R techniquess For these (1) See, for instance: Griffith, J. S. In Theory of Transition Metal Ions; Cambridge University Press: New York, 1961. Chien, J. C. W. J . Chem. Phys. 1969, 51, 4220. Montgolfier, P. D.; Harriman, J. E. J. Chem. Phys. 1971,55, 5262. Peisach, J.; Blumberg, W. E.; Adler, A. Ann. N.Y. Acad. Sci. 1973,206, 310. Weissbluth, M. In Hemoglobin; Springer-Verlag: New York, 1974. Taylor, C. P. S . Biochim. Biophys. Acta 1977, 491, 137. (2) Griffith, J. S. In ref 1. Weissbluth, M. In ref 1. Taylor, C. P. S. In ref 1. Rynard, D.; Lang, G.; Spartalian, K.; Yonetani, T. J. Chem. Phys. 1979, 71, 3715. (3) Pryce, M. H. L. Proc. R . SOC.London, A 1951, 205, 135. LonguetHiggins, H. C.; Stone, A. J. Mol. Phys. 1962, 5, 417. Carrington, A.; McLachlan, D. A. In Introduction to Magnetic Resonance; Harper and Row: New York, 1963. Stone, A. J. Proc. Roy. SOC.London, A 1963, 271, 424. Tippins, H. H. Phys. Reu. 1967, 160, 343. Atkins, P. W.; Jamieson, A. M. Mol. Phys. 1967, 14, 425. McWeeny, R.; Sutcliffe, B. T. In Methods of Molecular Quantum Mechanics; Academic: New York, 1969. Moreno, M. J. Phys. C: Solid State Phys. 1976, 9, 3277. de Brouckere, G.; Trappeniers, N. J.; ten Seldam, C. A. Physica B+C (Amsterdam) 1976,84,295. Hegstron, R. A. Phys. Rev. A: Gen. Phys. 1979,19, 17. Ishii, N.; Kumeda, M.; Shimizu, T. Solid State Commun. 1982, 41, 143. (4) (a) Feher, G. Phys. Rev. 1956, 103, 500. Scholes, C . P.; Isaacson, R. A.; Feher, G. Biochim. Biophys. Acta 1972, 263, 448. Van Camp, H. L.; Scholes, C. P.; Mulks, C. F. J. Am. Chem. SOC.1976, 98,4094. (b) Scholes, C. P.; Van Camp, H. L. Biochim. Biophys. Acta 1976, 434, 290. Scholes, C. P.; Falkowski, K. M.; Chen, S.; Bank, J. J. Am. Chem. SOC.1986, 108, 1660.
0022-3654/89/2093-0194%01.50/0
nuclei, ab initio molecular orbital wave functions obtained by the self-consistent charge extended Huckel (SCCEH) procedure6 have provided successful explanations' of the observed hyperfine interactions in a number of high-* and low-sping derivatives. We have therefore been interested in examining how well the calculated electronic structures from SCCEH investigation, which have successfully explained' hyperfine data, can also explain observed results of g-tensor measurements. In this respect, the analysis of the g tensor provides an added dimension to the examination of the accuracy of the calculated electronic structure, because it involves the wave functions and energy levels for excited states'-3J0 in addition to those of the occupied states, which are needed for the interpretation of the hyperfine properties. In earlier worklo we used conventional perturbation theory to study the departure of the g tensor from isotropic character with value g = 2.0023 due to the combined influence of the spin-orbit and orbit-magnetic field interactions. In this type of perturbation treatment, one encounters problems when the excitation energy from an occupied state to an excited state is comparable to the spin-orbit interaction. This was the case in our earlier worklo for ferricytochrome c, where the separation between the doubly occupied d,,-like and singly occupied d,,,-like levels was comparable to the spin-orbit coupling parameter for iron. This requires one to use higher order perturbation theory, which is rather cumbersome and was treated in an approximate manner in our earlier work.'O In the present work we deal with the influence of the spin-orbit interaction through a variational procedure using the calculated molecular orbital energy levels and wave functions of low-spin hemoglobin derivatives. The variational procedure has the attractive feature of allowing one to include spin-orbit effects to all orders, thereby permitting in principle greater accuracy than ( 5 ) La Mar, G. N.; Ropp, J. P. Biochem. Biophys. Res. Commun. 1979, 90, 36. Nagai, K.; La Mar, G. N.; Jue, T.; Bum, H. F. Biochemistry 1982, 21, 842. Takahashi, S.; Allison, K. L.; Lin, C.; Ho, C. Biophys. J. 1982.39, 33. (6) See, for instance: Hoffman, R. J. Chem. Phys. 1963, C9, 1397. Zerner, M.; Gouterman, M.; Kobayashi, H. Theoret. Chim. Acta 1966,6,363. Han, P. S.; Das, T. P.; Rettig, M. F. Theoret. Chim. Acta 1970, 16, 1. Trautwein, A,; Zimmermann, R.; Harris, F. Theoret. Chim. Acta 1975, 37, 89. Loew, G.;Kirchner, R. F. J. Am. Chem. SOC.1975, 97, 7388. Mishra, K. C.; Mishra, S. K.; Das, T. P. J. Am. Chem. SOC.1983, 105, 7729. (7) Mallick, M. K.; Chang, J. C.; Das, T. P. J. Chem. Phys. 1978, 68, 1462. Mun, S. K.; Mallick, M. K.; Mishra, S.; Chang, J. C.; Das, T. P. J. Am. Chem. SOC.1981, 103, 5024. Mishra, K.C., et al., In ref 6 . (8) References 4a and 5. (9) Reference 4b. (10) Mishra, K. C.; Mishra, S. K.; Roy, J. N.; Ahmad, S.; Das, T. P. J . Am. Chem. SOC.1985, 107, 7898. (1 1) Montgolfier, P. D.; Harriman, J. E. In ref 1.
0 1989 American Chemical Society
EPR of Ferricytochrome c and Azidomyoglobin
The Journal of Physical Chemistry, Vol. 93, No. 1, 1989 195
in the conventional perturbation approach. It also avoids the special problems that arise when one has closely lying levels to deal with. The systems we have studied here are ferricytochrome c and azidomyoglobin, both because there are accurate experimental data on these systems and also for comparison with earlier results1° using perturbation theory. Section I1 will briefly deal with the theory and procedure used in our work. In section 111, we shall present the results of our investigation and discussions. In section IV, we shall present the conclusion from our work and discuss the possible directions for further improving the agreement between theory and experiment, especially additional mechanisms that could contribute to the g tensor that have not been explicitly considered before.
+ t + -4-
-$+-++-
-+I-
I
1
,
I
**
s,*, =
(1)
f!/2**
-*:-
-+++I
11. Theory and Procedure
As discussed in perturbation approaches to the g tensor and in particular in our earlier work,1° in a spin l/z system, the states Q,1/2 corresponding to quantum numbers *1/2 for the Z component, S, of the total spin get admixed with each other through the spin-orbit interaction. This mixing is also accompanied by unquenching of the orbital angular momentum of the paramagnetic ion, iron in our case. Consequently, the admixed states are no longer eigenfunctions of the z component, S,, of the total spin of thg system. However, one can define12 an effective spin operator S for which
-*+I-
.r
Figure 1. Schematic representation for the ground-state occupancies of one-electron levels corresponding to (left) M,= and (right) M,=
where solid lines represent five d-like molecular orbital states and broken lines represent states with partial d- or ligand-like characters. orbitals of the system. The i A in eq 7 refers to the orbital angular momentum vector about the nucleus of the atom A. The eigenfunctions 9,of % in eq 3 will be obtained by a variational procedure involving the expansion
The g factor for such a system is in general tensorial in character, leading to the spin Hamiltonianl2 --:
%, = pgH*p'S
(2)
where pg is the Bohr magneton. In the present work, the eiin eq 1 will be obtained by a variational procedure genfunctions involving the diagonalization of the Hamiltonian: % = %yo %so (3)
*,
+
In eq 3
with ho(i) representing the one-electron Hamiltonian for the ith electron given by ho(i) = -(h2/2m)V:
+ Vo(i)
i
4 and correspond to determinantal functions, involving different occupations of the various one electron levels in Figure 1. The lowest two basis states *o,+1/2 and @o,-l/2 in the summation in eq 8 correspond respectively to the occupancies shown in Figure 1. On applying the usual variational procedure involving the minimization of the expectation value (*,l%l*,) with respect to the variational parameters Di,y&,, one obtains the following linear equations:
where
(5)
the first term being the kinetic energy operator and the second term the one-electron potential experienced by the ith electron. The form for Vo(i) depends on the approximation used. Thus, in Hartree-Fock theory,13this one-electron potential involves the sum of the Coulomb and exchange interactions of the ith electron with other electrons in the system. In the SCCEH theory: one does not need an explicit form for Vo(i), since one relates the matrix elements of ho(i) to the ionization energies between electronic orbitals on different atoms. For the spin-orbit Hamiltonian, we shall employ the conventional approximate form used in the literature,lZ namely
%so = E h s o W
In eq 8, the basis functions *y+M, are eigenfunctions of 7foin eq
7fy,M,;y',M,
sy,M,;yf,Mf,
= ( *y,M,l*y',M's)
=
6yy'8M&f1a
(10)
(11)
being unity for a = b and zero for a # b. Equation 11 follows are orthonormal. because the many-electron functions Equation 9 leads to the determinantal equation equation
*,,
det
- 6yy'6M,M',El = 0
I%fy.M,;y',M',
(12)
The matrix elements %y,M,;r',Mt, can be shown14 by using eq 3-7 to be given by the expressions %y,M';y,M8
(6)
= E(4,lho + hSOl4J c
(13)
for the diagonal elements and %y,M,;y',M',
Z{AiA*?
(*y,M,I%l*y~,M~J
aab
where hso =
=
(7)
= *($plhO + hSO14cf)
(14)
B. In Electron Paramagnetic Resonance of Paramagnetic Ions;Clarendon:
for the nondiagonal, where the 4,, represent one-electron molecular orbitals corresponding to the various energy levels in Figure 1. In eq 14, because of the one-electron character of ho and hso, the only nonvanishing matrix elements are14 those for which the differ by many-electron states corresponding to *y,M, and 97tM, the occupancy of only a pair of states corresponding to @, and If the many-electron states differ by the occupancies of more than a pair of states, then %y,M,;y'Ms vanishes. The plus or minus
Oxford, UK, 1970. (13) See,for instance: R a d , B. J. Rev. Mod. Phys. 1960, 32, 239, 245. Moskowitz, J. W.; Snyder, L. C. In Merhods of Electronic Structure Theory; Plenum: New York, 1977.
1978.
A
In eq 7,{A is the spin-orbit parameter of the atom corresponding to the particular atomic orbital component under consideration in the matrix element expression for hso involving the molecular (12) Carrington, A.; McLachlan, D. A. In ref 3. Abragam, A.; Bleaney,
(14) Weissbluth, M.In Atoms and Molecules; Academic: New York,
196 The Journal of Physical Chemistry, Vol. 93, No. 1, 1989
signs occurs when the number of permutations required to move from the row in the determinantal function \k,,M,corresponding in the determinantal function Q,r,Mf, is even to +,, to that for or odd, respectively. With use of eq 7 for hso and the fact that the 4, are eigenfunctions of ho in eq 5, so that
hO4# = e,+,
(15)
eq 13 and 14 reduce to 7fy,M,;,.M,
= Ce, = E
Roy et al.
(*llCH,C(l,i + gGyi)I*l) (22b) v
i
where y runs over X , Y,and Z . In determining the tensor components g,a by equating both sides of eq 22a and 22b, one has to make use of the relations
(16)
II
~y,,w,;y~,,w, = (+,IC~A~AW,O A
(17)
In eq 17, the suffix A has to be applied to the atomic orbital component corresponding to atom A in the molecular orbital 4,. Thus (dpl
A
lA7A*s14p')
=
ij,A
c~,cfi'jcA
(XiAIIA'qXjA)
( 18)
where the molecular orbital function 4, is given by6*Io
4, =
CCFixi i
which follow since S represents the effective spin of the system, the first two relations in eq 23 being the same as in eq 1. One thus obtains
(19)
xi being the atomic orbitals used as a basis set, and the atomic orbital xiA in eq 18 is assumed to be on atom A. In eq 18 if there is more than one orbital on atom A, such as 3d and 4p states on iron atom, one has of course to use different spin-orbit constants for the different atomic orbitals. Because of the short-range nature of the spin-orbit intera~tion,'~ one has to include only the terms in the molecular orbital matrix elements in which the atomic orbitals xi and xi are both on the same atom A as has been done in eq 18. With the Hamiltonian matrix elements constructed in this manner, the secular equation in eq 12 is solved to determine the energy eigenvalues E . These are in turn used in the conventional manner16 for the variational procedure to obtain the Di,,,+,, in eq 9 by using besides these equations the normalization condition
gyz = 2 Re
(qIIC(1yi i
+ g$yi)IQl)
(23c)
which yield the nine components gyp with y,d = x , y , and z. It should be noted that unlike the situation in the perturbation approach,I0 the components of the g tensor in the chosen axis system as given by eq 23a-c are not necessarily symmetric. However, as has been discussed in the literature,2 the diagonal components of the g tensor that are obtained experimentally are given by
g+,, = (Gytyl)1/2
(24)
where Gyq is obtained by diagonalizing the G tensor in the chosen axis system with components given by The molecular orbital wave functions in eq 19 used in the present calculations were obtained by the SCCEH procedure6 and have been used in our earlier investigation^^^*'^^'^ of the g tensor and hyperfine properties of ferricytochrome c and azidomyoglobin. Once the Di,y,M,are determined, one obtains from eq 8 the eigenfunctions \ki of the molecular Hamiltonian, including the spin-orbit interaction. The two lowest eigenfunctions Q, and \k2 are degenerate and form the lowest Kramer's doublet,I9 corresponding to 9+and Q-, respectively, in eq 1. An applied magnetic field removes this degeneracy and leads to zeeman split levels, the transition between which corresponds to the conventional EPR frequency. The Zeeman Hamiltonian involving the interaction of an applied magnetic field with the orbital and spin angular momenta of the individual electrons is given byI2
7fz = F B G * j ' c ( l j A i
+ g$j)
(21)
where g, is the free-electron g factor 2.0023. To obtain the components of the tensor g in eq 2 for the spin Hamiltonian, one has to equate the matrix elements of %, and 7fzover \kl and \k2; thus
= ~g-ibgsb
(25)
where g.,b and gab are obtained from eq 23a-c. The tensor G,, is symmetric and can be diagonalized. In the perturbation approach, which we have used earlier,1° the g tensor in the chosen axis system was symmetric and was directly diagonalized to obtain the principal components to compare with experiment. However, it should be mentioned here that if the g tensor is symmetric as is the case in the perturbation approach, the principal components obtained from diagonalizing this tensor are the same as those obtained from diagonalizing the G tensor. This can be shown as follows: Thus if
gjk = gk] then
Cgjgkb = Cgjgbk = 6
b
(&jk
In the principal axes system where G is diagonal, one can then = (gi.i.)2,g f f being the ith principal component write Grit = of the g tensor, the last step following because the g tensor is now diagonal. 111. Results and Discussion
(1 5 ) Watson, R.E.; Freeman, A. J. In Hyperfine Interactions; Academic: .. . New York, 1967. (16) Schiff, L. I. In Quantum Mechanics; McGraw-Hill: New York, 1968. (171 Mishra. K. C. et al. In ref 6. Mishra, S . K.: Roy. J. N.: Mishra, K. C.; Das, T. P., to be submitted for publication. (18) Mishra, S. K. et al. In ref 17. Roy, J. N . In Ph.D. Thesis, State University of New York at Albany, Albany, NY, 1987. (19) Weissbluth, M. In ref 1 .
Ferricytochrome c. We shall first present and discuss our results for ferricytochrome c. The model system chosen for this molecule is the same as in our earlier investigations on this system and is shown in Figure 2, the iron atom lying on the porphyrin plane. To explain the nature of the determinantal wave functions \k,,M8 in eq 8 used for the trial function for this system, it is necessary to refer to Figure 1, where the d-like molecular orbitals have been shown by solid lines and their populations in the ground state are
EPR of Ferricytochrome c and Azidomyoglobin
The Journal of Physical Chemistry, Vol. 93, No. 1, 1989 197 used. For the occupied states, as in the case of the determinants involving d-like states above, excitations of the down spins to the dYzstate were included, while for the empty states, excitations of the up spin from the d,, state were considered. The 25 determinants for M, = -lI2 were again obtained in an analogous manner, reversing the roles of up and down spins. The results that we have obtained using the procedure described i_n section I1 for the g tensors given by eq 23a, 23b, and 23c and G given by eq 25 and the principal components gqy of the g tensor defined by eq 24 are shown in eq 26-28 and 29-3 1. The results are given for the two cases involving 10 and 50 determinants qTfiS in eq 8. Ten-Determinant Case: g =
G= H34
I
H33
1.793 -0.063 -0.030
-0.107 2.326 0.064
-0.063 0.112 3.126
3.229 -0.369 -0.258
-0.369 5.428 0.501
-0.258 0.501 9.778
( ( (
gdd = 1.778, gff = 2.328, Figure 2. Model system for ferricytwhrome c, where hatched circles A
and B represent methionine-91 and imidazole ligands, respectively, and X and Y axes are passing through methionine carbons C6,CI1,C16,and C21. The other carbons of the pyrrole rings are represented just by numbers, whereas the side chains of the porphyrin ring are replaced by hydrogen saturators. ENERGY 1 ev)
-8.0929
-8.1956
-1
-I
-
1.2224 -
1,1898
-113391
WAV E
LEVEL
-
-.e02 I d Z 2
t 01 6 nd,
gztd = 3.138
Fifty-Determinant Case: g =
1.758 0.144 0.103 2.444 0.050 -0.021
0.132 -0.053 3.165
- F U Y C TIC. N +.0c2 I dX2-y2
- 01 41d y z --.OI 6 4 d K y + T r . '
G=
- . C l ? l d z z + , 0 0 0 2 d X Z ~ ~ O o O O d x 2 ~ ~ 2 t ~ C 0 6 5 d y ~2dxy + . 8 +I(..) Cl
- . 0 4 3 2 d Z ~ + . I5 9 2 d X Z - . 0 0 2 ? d ~ < ~ t2 8 0 4 3 d y Z ~ c 0 3 6 d X y t I i ~ ~ ~
.O329dZ2+.8988dXZ t .
c o ~ o d ~ z+ .~0 ,7 ~6 8 d y Z -.ooo8dXy tri..)
. c l 1 2 d Z 2 - ~ O c 6 0 d X Z + . 9 7 5 ? d X 2 ~ y 2.$C 1 6 2 d y Z + . 0 0 2 0 d X y + T ~ . . l
Figure 3. Level energies and wave functions for the five d-like molecular orbitals in the ferricytochrome c system corresponding to ground-state configurations in Figure 1 (shown by solid lines).
shown for the two cases where the net spins are up and down (Figure 1, left and right, respectively), that is, M, = +l/z and We have carried out two investigations for this molecule. In the first, 10 determinantal functions were used in eq 8. The first two determinants refer to the occupancies in Figure 1 with eight others obtained from excitations within five d-like orbitals which are related to these two ground states. The energy levels and wave functions for the five d-like orbitals (indicated by solid lines in Figure 1) are shown in Figure 3. Two of the excited many electron states were obtained from Figure 1, left, by exciting the down-spin electrons in the d A 9 - and d,-like states to the d,,-like state. Two others were obtained by exciting the up-spin electron in the d,,-like state to the empty d, and dz-like states. The other four excited states were exact counterparts of those obtained from excitations associated with Figure 1, right, by interchanging the roles of up and down spins. In the second investigation, to study the influence of other states besides the d-like states, we have utilized for the construction of a number of additional the determinantal wave functions q,,M, one-electron molecular orbital states, for which the coefficients of the d-orbitals were 0.1 or higher. A set of 50 determinants were constructed from these and the d-like orbitals in Figure 1, with 25 corresponding to M, = +'I2and an equal number for M, = - I / * . For the construction of the 25 determinants with M , = 20 other one-electron states (both doubly occupied ones and empty ones) besides the d-like states shown in Figure 1, left, were
(3.129 0.526 0.501
gdd = 1.731,
0.526 5.988 -0.212
gff = 2.470,
1
0.501 -0.212 10.018
) )
gztz,= 3.172
On comparing the g tensor components in eq 26 and 29, we find that both the diagonal and nondiagonal components are in general affected in the second decimal place in going from the 10-determinant to the SO-determinant case. Percentagewise, the off-diagonal elements appear to be more severely affected because of their-relatively small sizes. The differences in the two cases for the tensor are further accentuated because of the bilinear nature of the expression in eq 25. As regards the principal components of the g tensor that can be compared with experimental results, one notices that the differences between 10- and 50-determinant cases are much less pronounced than for the g and G tensors for the molecule-based axis system, most likely because the off-diagonal components influence the principal components much less sensitively than do the diagonal ones. However, the differences are quite significant and more so when one considers the g shift, that is, the difference between the principal components and the free-electron g value, 2.0023. These results indicate that while the differences in the results between 10- and 50-determinant cases are not so large, for accurate quantitative analyses of the g tensors, it is desirable to take account of other states besides the d-like states alone. The orientations of the principal axes (X', Y', 2') with respect to the molecule-based axes X,Y, and Z are given in eq 32 for the 50-determinant case. The columns denoted
c
0.938 0.180 0.030
-0.178 0.981 -0.075
-0.043
0.069 0.997
by i'represent principal axes X',Y', and Z', and the rows denoted by i refer to molecule-based axes X,Y, and 2. In Table I, we have compared the principal components of the g tensor from the present investigations with the results obtained
198 The Journal of Physical Chemistry, Vol. 93, No. 1, 1989
TABLE I: g Values for Ferricytochrome c Obtained from Perturbation and Variational Approaches and Comparison with Experimental Values
perturbation R
U
b
variational
gyy
2.010 2.449
2.009 2,356
2.470
g,
1.730
exptl (EPR)‘ 1.250 2.250
3.170 3.060 3.037 3.716 First-order spin-orbit effect included. *Higher order spin-orbit effects are esdmated approximately. Reference 20. gf+
Figure 4. Orientations of the principal axes in the ferricytochrome c
system obtained from variational and perturbation calculations and comparison with experimental predictions. earlierI0 by perturbation theory to first order in spin-orbit interaction and including approximate estimates of higher order effects as well as with the experimental values obtained from single-crystal EPR measurements.zo One of the features that emerges from this comparison is the closeness in the values of the largest component obtained by the perturbation procedurelo by using estimates of higher order spin-orbit effects and variational methods with experiment2’ In the earlier work by perturbation method,1° it had been found that this particular component was rather sensitively dependent on the small separation between the dx,- and d,,-like levels. One has to include a sizeable correction to this spacing due to the influence of higher order spin-orbit effects to arrive at the perturbation result listed in column three of Table I. The fact that the variational result is in good agreement with the perturbation value after the correction to the spacing of the d, and dyzlevels has been included indicates that the variational procedure satisfactorily includes the influence of higher order spin-orbit effects. The other notable feature from Table I is that the smallest component from the perturbation approach was close to, and actually slightly higher than, the free-electron g value, while the experimental value was substantially smaller. The variational procedure leads to a value for gdd considerably smaller than the free-electron value although there is still some significant difference with experiment. The values of gvv from perturbation and variational investigations differ relatively less from each other as compared to the situation for gztzt and in fact the variational result is in somewhat lesser agreement with experiment as compared to the perturbation result. The orientations of the principal axes from variational and perturbation calculations and experiment are shown in Figure 4. It can be seen that both methods lead to good agreement with experiment20 for the Z’axis. The X’and Y’axes lie close to the ~
~~
(20) Mailer, C.; Taylor,
C. P. S. Can. J . Biochem.
1972, 50, 1048
Roy et al. heme plane. However, there seem to be significant differences in their orientations as compared to experiment, the angles between the directions from variational calculations and experiment being about 30° and 40’ for X’ and Y’axes, the difference being somewhat smaller as compared to that for the perturbation results.’O In attempting to understand the remaining differences between experimental and theoretical components of the g tensor in Table I and the orientations of the principal axes in Figure 4, one has to consider several possible sources that could contribute to these differences. An obvious possibility is the need for more accurate energy levels and electronic wave functions than those used in the present work and the earlier perturbation treatment.lo The energy levels and wave functions in the present work have been obtained by the SCCEH procedure6 by using a semiempirical treatment of the matrix elements of the Hamiltonian. This was necessitated by the large size of the molecular system that we have to work with. To obtain more accurate energy levels and wave functions, it will be preferable to use the Hartree-Fock procedure.21 For a molecule of the size of the model system we have chosen for ferricytochrome c, this appears to be beyond the capabilities of conventional high-speed computers, and one would have to use a supercomputer. However, in addition to the need for more accurate electronic structure, there could be some other mechanisms that could contribute to the g tensor that we have not included in our current investigation. One such mechanism is the exchange polarization effect,22which has been found to be important for hyperfine properties and arises from the difference between the electronic wave functions for paired spin states of different spins (including core states) due to the different exchange interactions experienced by electrons in different spin states in a paramagnetic molecule. This difference would allow the paired spin states to make finite contributions to the g tensor, as has been pointed out in earlier worklo involving the perturbation approach. To include this exchange polarization affect, one would have to use an unrestricted Hartree-Fock (UHF) approach23or some other approximation to the U H F procedure. Additionally, there are perturbation method^^^,^^ that have been applied in the literature for the study of exchange polarization contribution to hyperfine interaction properties when dealing with restricted Hartree-Fock wave functions, and these could be adapted to the g-tensor problem. All these improvements to the procedure used here would be rather time consuming but should be attempted in future. Azidomyoglobin. The model system chosen for this molecule is shown in Figure 5 with the iron atom on the porphyrin plane. This model has also been used1* for the investigations of the hyperfine properties of this system. The plane of the porphyrin was taken as the XY plane with the X and Y axes passing through N2-N4 and N2-N5 atoms, respectively. The fifth ligand was a protonated imidazole, its plane being taken as that formed by the Z axis and a line on the X Y plane inclined at 61 O to the X axis as shown in Figure 5 based on available structural data.2s The azide group (N3-) forms the sixth ligand, being linear and inclined (21) Hartree, D. R.; Hartree, W. Proc. Roy. Soc. London, A 1948, 193, 299. Roothaan, C. C. J. Rev. Mod. Phys. 1951, 23, 69. Pople, J. A,; Nesbet, R. K. J . Chem. Phys. 1954.22, 571. Bagus, P. S.; Liu, B. Phys. Rev. 1966, 148, 79. Dedieu, A.; Rohmer, M. M.; Veillard, A. Proceeding of the 9th Jerusalem Symposium on Quantum Chemistry and Biochemistry, March, 1976. Dedieu, A.; Rohmer, M. M.; Bernard, M.; Veillard, A. J . Am. Chem. SOC.1976, 98, 3717. Messmer, R. P.; Knudson, S. K.; Johnson, K. H.; Diamond, J. B.; Yang, C. Y. Phys. Rev. B: Condens. Matter 1976,13, 1396. Huynh, B. H.; Case, D. A.; Karplus, M. J . Am. Chem. Soc. 1977, 99,6103. Sahoo, N.; Mishra, S. K.; Mishra, K. C.; Coker, A,; Das, T. P.; Mitra, C. K.; Snyder, L. C.; Glodeanu, A. Phys. Rev. Lett. 1983, 50, 913. Mohapatra, S.; Sahoo, N.; Dev, B. N.; Mishra, K. C.; Gibson, W. M.; Das, T. P. J . VUC.Sci. Technol. 1986, A4(6), 2441. (22) See, for instance: Gaspari, G. D.; Shyu, W.; Das, T. P. Phys. Reu. A: Gen. Phys. 1964, 134, 852. (23) Watson, R. E.; Freeman, A. J. in ref 15. Bagus, P. S.; Liu, B. In ref 21. Ray, S. N.; Lee, T.; Das, T. P.; Phys. Rev. B Condens. Matter 1973, 8, 5291. Rodgers, J. E.; Das, T. P.;Phys. Rev. A: Gen. Phys. 1973,8, 2195. (24) Ikenberry, D.; Rao, B. K.; Mahanti, S. D.; Das, T. P. J. M a p . Reson. 1969, 1 , 221. Ikenberry, D.; Jette, A. N.; Das, T. P. Phys. Rev. B: Condens. Matter 1970, 1 , 2785.(25) Stryer, L.; Kendrew, J. C.; Watson, H. C. J. Mol. Biol. 1964,8, 96. Dickinson, L. C.; Chien, J. C. W. J . Biol. Chem. 1977, 252, 1327.
EPR of Ferricytochrome c and Azidomyoglobin
The Journal of Physical Chemistry, Vol. 93, No. 1, 1989 199
IY
to the Z axis2’ a t 69O. The azide group and the heme normal form a plane that passes through two methionine carbon atoms opposite to each other as shown in Figure 5 and makes an angle of 64O with respect to the plane of the imidazole. The N, atom of the azide group is located a t a distance of 2.1 A from the Fe atom, the separation between successive nitrogen atoms of N3group being 1.15 A. One important difference in the choice of X and Y axes in this case as compared to the ferricytochrome c system is that these axes now point along pyrrole nitrogens rather than along the meso carbons. A consequence of this is that in Figure 6, representing the energy levels and eigenfunctions for the five d-like levels in this molecule, the lowest level has d,,-like symmetry in contrast to d,+g in ferricytochrome c, which is the more usual convention in the literature. Another consequence of the present choice of X and Y axes is that now there is a strong admixture of d,- and d,-like states in the molecular orbital (MO) wave functions corresponding to the two states associated with these symmetries. The variational functions represented by eq 8 were again constructed in exactly thg same manner as for ferricytochrome c and calculations of the G tensor were carried out with both 10and 50-determinantal functions. The g tensor obtained in the two cases are given in eq 33 and 36, respectively. The corresponding Ten-Determinant Case: 2.162 g = (0.303
-0.007
G=
(
4.763 1.308 -0.032
0.301 2.171 -0.008
:::3: -0.036
-0.005 -0.007) 2.555
4:;CLi) 6.529
Figure 5. Model system for azidomyoglobin, where orientations of the imidazole plane as well as the plane of the azide group have been shown by broken lines, and X and Y axes are shown by solid lines passing through Nz-N, and N,-N5 nitrogens of the pyrrole rings, respectively. ENERGY(eV1
- 8 I579 -92768
gdy
= 1.864, g,,/ = 2.467, gzty = 2.556
Fifty-Determinant Case:
WAVE - FUNCTION
LEVEL ~
--
0000dz2+0018d,z * 8088dX2.y2-0005dyz-WOOdxy+ 7813dz2-0244dxz + 0000dx2.y2 0248dyz-0041dxy
I( I
-I
(
- 1 1 I343 -- 0007dz2-5645d,2 -0001 dx2.y2- 5730dy2-0002dxy* 1 ( - 1 1 1454 - - - - - - -0419d,2-1366d,z +0001dxZ.y2- 0934dyz-0009d,y* Z ( - 1 I2363 --0152dz2-6390d,z + 0004d~2.~2-6392dyz+ 0320dxy+ I I - 1 1 2639 --0009dz~*0205dxz-0000d&y2+ 0204dyz+9832d,y* I I - 1 1 7755 - - - - - - - - 1784dZZ+0929d,z - 0003dx2 y 2 + 0958dyz 0012dxy+ I (
) )
-124189 - - - - - - 0228dz2-2137d,~+ 0019dX2.y2- 2 0 5 0 d y ~ - 0 0 0 8 d ~ yI+( -126483 - - - - - - - -0002dz2+0007dx~- 4 1 3 7 d l ~ y 2 ~ 0 0 1 6 d y z + 0 0 0 0 d , I y ~(
1
+
g=
-0.023 G = (4.905 1.512
-0.107 gyd
:S:i
4:&;)
-0.107
7.167
-0.024
2.677
= 1.843, gY , = 2.528, gz,zt= 2.683
symmetric C tensors, defined in eq 25, are given in eq 34 and 37, respectively. The principal components gdy, g,, andgii that are obtained through diagonalization of the symmetric C tensors by using eq 24 are presented in eq 35 and 38, respectively. Equations 33 and 36 show changes in general in the second decimal place in all the components in going from the 10-determinant to the 50-determinant case. The off-diagonal elements are percentagewise more different for the two basis sets than are the diagonal components. Also, the components of the G tensor undergo stronger changes than the g tensor in going from the lbdeterminant case to the 50-determinant case. All these features are similar to those found in the case of ferricytochrome c. However all these differences are somewhat smaller in magnitude than in the case of ferricytochrome c. The principal components of the gdd, gyy, and gyy are seen from eq 35 and 38 to differ much less significantly from each other as compared to the cases of the nondiagonal g tensor and the symmetric G tensor due to the same reason as given in section 111 for ferricytochrome c. However, there are still some differences between the results for the two basis sets, those for the case of the larger set being in somewhat better agreement with experiment as seen from Table 11.
1
I I I 1
Figure 6. Level energies and wave functions for the five d-like molecular orbital states (shown by solid lines) as well as a few other states (shown by broken lines) adjacent to these five d-like states corresponding to the ground-state configuration in the azidomyoglobin system. TABLE II: g Values for Azidomyoglobin from Perturbation and Variational A p p r o a c h and Comparison with Two Sets of Experimental Single-Crystal EPR Results
EPR
perturbation variational a b 1.843 1.720 1.720 gA3.J 1.946 2.528 2.190 2.220 gw 2.514 2.683 2.820 2.800 gii 2.725 ‘Data are obtained from ref 26a. *Data are obtained from ref 26b. g
Therefore, as remarked for the case of ferricytochrome c, for accurate quantitative comparison between theory and experiment it is desirable to use the larger basis set, which includes excitations outside of the manifold of the d-like molecular orbital states. The orientations of the principal axes with respect to the molecule-based axes X , Y, and Z are given in eq 39 with the same convention for the matrix elements in this equation as in the eq 32 for the ferricytochrome c case. 0.708
0.694
-0.135
0.000
0.191
0.982
(39)
200
The Journal of Physical Chemistry, Vol. 93, No. 1, 1989
In Table 11, the principal components gdd, guy, and gii from the perturbation and variational (for the 50-determinant case) calculations are compared with the experimental results from polycrystalline and single-crystal EPR measurements.26 The variational results are seen to be in better overall agreement with experiment than the perturbation results. The most significant difference between perturbation and variational results appears to occur for the smallest g component, gdd, although the difference is less pronounced than for ferricytochrome c. The variational result is again closer to experiment than the perturbation result. For the largest component, gfi, the variational and perturbation results are close to each other and to the experimental results. On the other hand, for guy, while the variation and perturbation results are close to each other, they are both significantly larger than experiment. Next, considering the orientation of the principal axes, the directions of the X’and Z’axes from the perturbation and variational approaches and experiment are shown with respect to the molecule-based coordinate axes X, Y, and Z (Figure 5 ) in Figure 7 . In obtaining these directions, we have nlade use of -yir for the variational case from eq 39, while those for the perturbation theoretic approach are taken from our earlier work.I0 The Y’axis is mutually perpendicular to X’and 2’and has not been shown in Figure 7. From Figure 7 , it can be seen that the angles between the Z’and 2 axes for the perturbation and variational approaches differ quite significantly, from 23’ in the former case to 1 1O in the latter, the variational results being closer to the angle of about 9’ found from single-crystal experimental measurements.26b As regards the X’axis corresponding to the smallest principal component, gyd,the angle between X‘and X i s found to be 4 5 O from both the variational and perturbation calculations. This angle appears to be bracketed between the two values 2 9 O and 5 9 O found for the same angle from two different measurements.26 There is no experimental result for the Y’direction, for which the angles with the Y axis for the perturbationlo and variational approaches are found to be quite close, namely, 4 9 O and 4 6 O , respectively. Thus, the variational results for Azidomyoglobin with experiment are found to be in better overall agreement with experiment% than those from the perturbation approach. However, there are still some significant differences between experiment and theory, although they are less pronounced than in the case of ferricytochrome c. Possible ways to improve agreement with experi(26) (a) Helck6, G. A.; Ingram, D. J. E.; Slade, E. F. h o c . R. SOC. London, B 1968,169,215. (b) Hori, H. Biochim. Biophys. Acta 1971, 251, 217.
Roy et al.
Figure 7. Orientations of the principal axes for the azidomyoglobin system obtained from perturbation and variational calculations and comparison with experimental predictions. Only the X’and Z’axes are shown, the Y’ axes for the different cases being perpendicular to the corresponding X’ and Z‘ axes.
ment, namely, utilization of more accurate procedures2’ for obtaining the electronic wave functions and energy levels and incorporation of exchange polarization e f f e c t ~have ~ ~ ,been ~ ~ discussed already in section 111 and should also be attempted in the future for azidomyoglobin. IV. Conclusion The variational procedure utilized in the present work for the calculation of the g tensor using molecular orbital wave functions and energy levels has been found to provide significant improvement in the agreement between theory and experiment for the principal components of the g tensor as compared to the perturbation theory approach employed earlier.I0 While there is good overall agreement between our theoretical results and experimental results from electron paramagnetic resonance measurements,20*26there remain significant differences in the magnitudes of some of the principal components, especially the smallest one in both molecules and in the orientation of two of the principal axes of ferricytochrome c. Possible sources for improving the agreement with experiment have been suggested, namely, use of wave functions from more accurate procedures2’ and incorporation of exchange polarization effects. Registry No. Ferricytochromec, 9007-43-6.
