Spin-Orbit and Spin-Polarization Effects in Metalloporphyrins

4 Spin-Orbit and Spin-Polarization Effects in Metalloporphyrins 1

Downloaded by UNIV OF CALIFORNIA SAN DIEGO on January 29, 2016 | http://pubs.acs.org Publication Date: October 15, 1986 | doi: 10.1021/bk-1986-0321.ch004

David A. Case

Department of Chemistry, University of California, Davis, CA 95616

Some general features of spin-orbit and spin-polarization effects in paramagnetic metalloporphyrins are outlined. Spin-orbit effects are modelled through a four-component relativistic molecular orbital formalism. Results for copper porphine show significant effects of unquenched orbital angular momentum on the Zeeman and hyperfine tensors at both the copper and nitrogen sites. Spin-unrestricted calculations on high-spin iron complexes that are models for the active site in deoxyhemoglobin illustrate the effects of exchange-correlation (polarization) forces. These split the spin-up and spin-down iron d-orbitals by 3-4 eV. I discuss how this splitting affects orbital populations and the identification of charge-transfer transitions in the near-infrared region. The use o f m o l e c u l a r o r b i t a l theory t o i n t e r p r e t the e l e c t r o n i c s t r u c t u r e and s p e c t r a of m e t a l l o p o r p h y r i n s has a l o n g and g e n e r a l l y s u c c e s s f u l h i s t o r y . ( 1 ) The e a r l y work was based on H u c k e l and extended H u c k e l models, w h i l e more r e c e n t l y c a l c u l a t i o n s employing o t h e r s e m i e m p i r i c a l o r ab i n i t i o t e c h n i q u e s have become f e a s i b l e . ( 2 ) For o p e n - s h e l l systems, two l i m i t a t i o n s o f the extended H u c k e l model become i m p o r t a n t even f o r q u a l i t a t i v e d i s c u s s i o n s . F i r s t , s p i n d i s t r i b u t i o n s a r e a l t e r e d i n i m p o r t a n t ways by exchange p o l a r i z a t i o n f o r c e s , w h i c h f a v o r c o n f i g u r a t i o n s i n w h i c h e l e c t r o n s o f the same s p i n are i n the same r e g i o n o f space. Second, s p i n - o r b i t c o u p l i n g mixes spin-up and spin-down c h a r a c t e r and a l l o w s f o r n o n - v a n i s h i n g e x p e c t a t i o n v a l u e s o f o r b i t a l a n g u l a r momentum. T h i s has i m p o r t a n t e f f e c t s on magnetic resonance p a r a m e t e r s , phosphorescence i n t e n s i t i e s , and so on. I n t h i s paper, I d i s c u s s one way i n w h i c h these e f f e c t s can be i n c l u d e d i n m o l e c u l a r o r b i t a l c a l c u l a t i o n s , u s i n g copper p o r p h i n e and the h i g h s p i n form o f i r o n p o r p h i n e as examples. 7

C u r r e n t address: Research Institute o f Scripps C l i n i c , Department o f M o l e c u l a r Biology, University o f C a l i f o r n i a at San D i e g o , L a J o l l a , C A 92037 0097-6156/ 86/ 0321 -0059S06.00/ 0 © 1986 A m e r i c a n C h e m i c a l Society

In Porphyrins; Gouterman, Martin, et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

PORPHYRINS: E X C I T E D STATES A N D DYNAMICS

60 Details of the Calculations

The Χα multiple scattering method generates approximate singledeterminant wavefunctions, i n which the non-local exchange i n t e r action of the Hartree-Fock method has been replaced by a l o c a l term, as i n the Thomas-Fermi-Dirac model. The o r b i t a l s are solutions of the one-electron d i f f e r e n t i a l equation (in atomic units) 2

[-(1/2) V + V

+ ν (+)]ψ.(+) = ε.(+)ψ.(+)

c

χα

(1)

Here ψ i s an o r b i t a l of an electron with M = l / 2 ( t ) , ε i s i t s oneelectron energy, V i s the c l a s s i c a l Coulomb p o t e n t i a l (including electron s e l f - i n t e r a c t i o n terms), and ν ^ represents the effects of electron exchange. In Slater's model, this i s related to ρΐ, the l o c a l density of electrons of the same spin 1/3 ν ( Φ ) = -3α(3ρ+/4π) (2) s


c

α

χ α

Here a i s an adjustable parameter, usually determined by comparing Hartree-Fock and Χα atomic c a l c u l a t i o n s . In the spin-unrestricted version, the spin-up and spin-down o r b i t a l s are d i s t i n c t , so that i n general the resulting wavefunction i s not a spin eigenfunction. These approximations, and additional ones involving the muffin t i n potential, have been described i n recent reviews. (3-4) In p a r t i c u l a r , the work reported here i s e n t i r e l y analogous to our e a r l i e r calculations on porphyrins,(5-10) where we discuss the method of calculation i n some d e t a i l . The r e l a t i v i s t i c (DSW) version incorporates the same approxi mations but starts from the Dirac rather than the Schroedinger wave equation,(11) (3) [ca«p + 6mc + (V + V )Ιι»]ψ = Νψ ~ ~ c χα where α and β are Dirac matrics, Ii» i s the 4x4 unit matrix, ψ i s now a four-component spinor, and the average of the spin-up and spin-down charge densities i s used i n calculating the exchange potential. The top two components of ψ correspond to spin-up and spin-down character, so that the molecular o r b i t a l s i n general w i l l be spin mixtures. Each o r b i t a l , however, as well as the o v e r a l l wavefunction, transforms according to the molecular double point group. Two new results are reported i n this paper. In the next sec tion, r e l a t i v i s t i c calculations on copper porphine are presented, that incorporate parameters i d e n t i c a l to those we have used earlier(5) for n o n r e l a t i v i s t i c c a l c u l a t i o n s . (This calculation assumes a planar porphyrin and uses a Cu-N bond distance of 1.98 Â ) . These are the f i r s t f u l l y r e l a t i v i s t i c molecular o r b i t a l calculations to be reported for a metalloporphyrin, or indeed for any molecule of this s i z e . The following section gives d e t a i l s of calculations on a high-spin form of iron porphine, using a geometry and parameters reported e a r l i e r . ( 7 ) This i s not the ground state for this molecule, but has some features relevant to the high-spin, f i v e coordinate structures found i n the deoxy forms of hemoglobin and myoglobin. 2

v


4.

CASE

Spin-Orbit and Spin-Polarization Effects in Metalloporphyrins

61

Spin-Orbit Effects i n Copper Porphine In many respects the results of a r e l a t i v i s t i c molecular o r b i t a l calculation on copper porphine are similar to the corresponding nonr e l a t i v i s t i c r e s u l t s . (5,_8) The porphyrin levels are v i r t u a l l y unaffected by spin-orbit mixing (e.g. the r a t i o of spin-down to spinup character i s always less than Π Γ ) , and the energy l e v e l spacings are unchanged, although a l l of the levels are higher i n energy by 0.5 eV than i n the corresponding n o n - r e l a t i v i s t i c calcula tion. The " c r y s t a l f i e l d " o r b i t a l s having s i g n i f i c a n t copper 3d character are more affected, and the r e l a t i v i s t i c results are given i n Table I. The o r b i t a l s are expressed i n a coordinate system i n which the methine carbons are on the χ and y_ axes. As required by r e l a t i v i s t i c symmetry, the o r b i t a l s are complex ( i . e . are expanded i n complex shperical harmonics Y £ ) and are mixtures of spin-up and spin-down character. In both r e l a t i v i s t i c and n o n - r e l a t i v i s t i c calculations there are four r e l a t i v e l y low-lying f i l l e d o r b i t a l s , formed from the d x - y , dxz, dyz, and d z atomic o r b i t a l s . Their energies range from -11.80 to -11.64 eV i n the n o n - r e l a t i v i s t i c calculation, and from -11.09 to -10.69 eV i n the present r e s u l t s . The f i f t h c r y s t a l f i e l d o r b i t a l i s p a r t i a l l y occupied and has about 60% dxy character (this o r b i t a l points toward the nitrogen atoms i n our coordinate system); i t s energy r i s e s from -8.95 to -8.09 eV upon incorporation of r e l a t i v i s t i c e f f e c t s . As a result of these changes, the copper d-levels are about 0.5 eV higher i n energy r e l a t i v e to the prophyrin π levels than was the case i n the n o n - r e l a t i v i s t i c calculation reported earlier. Table I also contains an analysis of the o r b i t a l character of these f i v e energy l e v e l s . These were determined from the fourcomponent spinors by neglecting the two lower, "small," components, and by assuming that the r a d i a l functions depend only upon i.e. that the r a d i a l functions for p i / 2 and P 3 / 2 , or for and ds/2> are the same. The o r b i t a l s may then be written i n " P a u l i " form as products of (complex) spherical harmonics and spin functions. Populations are equal to the squares of the absolute magnitudes of the c o e f f i c i e n t s l i s t e d i n Table I. [For a l l but 17e3 , an addi t i o n a l o r b i t a l (not shown) i s occupied which has the same energy but the opposite spin pattern ( i . e . α and 3 are interchanged).] It may be seen that there i s s i g n i f i c a n t spin mixing i n the lower four c r y s t a l f i e l d o r b i t a l s , e.g. that the 13e g o r b i t a l has 41% spin-up (a) character on copper (mostly i n d x - y = (Y22 + Y 2 - 2 ) / * ^ 2 ) , 45% dT[ character which i s spin-down (3), and smaller contributions of both spins from the porphyrin. Unlike [Pt(CN)w] ~ that we looked at earlier,(12) there i s no nearly pure d z ( Y 2 0 ) o r b i t a l , although most of this character i s i n the 1 3 e 2 o r b i t a l . The p a r t i a l l y occupied 17e3 o r b i t a l i s of greatest interest i n the interpretation of magnetic resonance parameters. As Table I shows, this o r b i t a l i s s p l i t o f f from the other copper _3d l e v e l s , and i s almost pure α spin, with only 0.14% population i n the spindown dTT atomic o r b i t a l . As written, the o r b i t a l i s nearly pure imaginary: for example, the c o e f f i c i e n t s of Y 2 2 and Y 2 - 2 are nearly equal and opposite, and (Y22 - Y2-2)/1^2 = dxy. This could be made into a r e a l o r b i t a l by m u l t i p l i c a t i o n by i , and i t should be


5

m

2

2

g

3

2

2

2

2

g

g


2

P O R P H Y R I N S : E X C I T E D STATES A N D D Y N A M I C S

62

T a b l e I . C r y s t a l f i e l d o r b i t a l s f o r Cu P o r state

energy, eV

13 3 g

-11.09

e

Pauli

decomposition

Cu:

-.460 Y2-2CX + .672 Y - i 3 - .444 Y22CI

N:

-.010 (1+i) Υ ι _ ι α + . 0 1 8 ( l + i ) Y i 3

2

0


+.028(l-i) Y

12

e

2g

-11.03

3

x l

a

Ca:

(.033 + .071i) Y i o 3

C3:

(.045 + .055i) Y i o 3 3

Cm:

.070i Y

1 0

Cu:

.117 Y

a - .347 Y

N:

0 0

2 0

a + .816 Y i 3 2

- . 0 1 6 ( l - i ) Υι-ια + . 0 1 6 ( l + i ) Y n a -.029(l-i) Yio3

Ca:

14

e

3g

-10.80

C(3:

(-.058 + .071i) Y i o 3

Cm:

.090i Y i o 3

Cu:

-.526 Y2-2CX - .590 Y - i 3 - .519 Υ 2 »

N:

13

6

2g

-10.69

-8.12

2

-.018(l+i)Y!-ia

(-.025 - .057i) Y i o 3

C3:

(-.038 - .047i) Y i o 3

Cm:

-.058i Yio3

Cu:

.281 Y o t - .865 Y 00

2 0

a - .343 Y i 3

(.013 - .032i) Y i o 3

C3:

(.022 - ,028i) Y i o 3

Cm:

-.034i Y

Cu:

-.543 Y - a + .587 Y 2

-.130i Y

0 0

1 0

3

2

.041i Y

0 0

2 2

a - .037 Y - i 3 2

a + .116(l+i) Y i _ i a

+ .1151(l-i) Ca:

2

- . 0 3 4 ( l - i ) Y i - i a + .034(l+i) Y n a

Ca:

N:

2

+ .023(l-i) Y n a

Ca:

N:

17

(-.045 + .093i) Y i o $

Yna

a - (.047 + .012i) Y i - i a

-(.047 - . O l l i ) Y n a C3:

-.021 Y

0 0

a + (.016 + .030i) Y ^ a

+(.016 - .030i) Y n a C o e f f i c i e n t s m u l t i p l y a n o r m a l i z e d r a d i a l f u n c t i o n s ( n o t shown), complex s p h e r i c a l harmonics Y^ , and s p i n f u n c t i o n s as i n d i c a t e d . V a l u e s f o r the l i g a n d a r e f o r a s i n g l e atom. C o e f f i c i e n t s s m a l l e r than 0.01 a r e n o t shown. m


4.

CASE


63

r e a l i z e d t h a t our c h o i c e of phase f a c t o r i s an a r b i t r a r y one w i t h i n each i r r e d u c i b l e r e p r e s e n t a t i o n . Hence the 1 7 e g o r b i t a l can be c o n s i d e r e d to be " a l m o s t " a t the n o n - r e l a t i v i s t i c l i m i t i n w h i c h a l l of the o r b i t a l s may be taken to be r e a l . The s m a l l d e v i a t i o n s from the n o n - r e l a t i v i s t i c symmetry, how e v e r , a r e q u i t e i m p o r t a n t f o r u n d e r s t a n d i n g the magnetic b e h a v i o r of CuPor. I t i s of i n t e r e s t to n o t e , f o r example, t h a t the c o e f f i c i e n t of the drr o r b i t a l (0.037) i s almost e x a c t l y e q u a l to the v a l u e of 0.035 t h a t one would o b t a i n from f i r s t - o r d e r p e r t u r b a t i o n t h e o r y , ( 1 3 ) λ/Δ, where λ i s the s p i n - o r b i t c o u p l i n g c o n s t a n t f o r Cu(ca. -830 cnT ) and Δ i s c r y s t a l f i e l d s p l i t t i n g between the dxy and drr o r b i t a l s (from T a b l e 1, 2.9 eV or 23,500 c m " ) . As a r e s u l t t h e r e are few s u r p r i s e s i n p r o p e r t i e s depending on the e l e c t r o n d i s t r i b u t i o n around c o p p e r — t h e r e s u l t s p r e s e n t e d below a r e i n good a c c o r d w i t h v a l u e s p r e d i c t e d from c o n v e n t i o n a l c r y s t a l f i e l d t h e o r y ( m o d i f i e d to i n c l u d e e f f e c t s of c o v a l e n c y ) . On the o t h e r hand, the n a t u r e of the s p i n - o r b i t m i x i n g i n the v i c i n i t y of the l i g a n d atoms i s much h a r d e r to u n d e r s t a n d s i n c e a l a r g e number of e x c i t e d s t a t e s can mix i n t o the ground s t a t e v i a s p i n - o r b i t c o u p l i n g , and i t i s a most d i f f i c u l t t a s k to account f o r a l l of them. I n f a c t , i t i s o f t e n assumed t h a t s p i n - o r b i t e f f e c t s a t f i r s t - r o w atoms may be i g n o r e d . Our r e c e n t DSW r e s u l t s f o r f l u o r i n e s bound to Xe and Np c a s t doubt upon t h i s assumption,(14-16) and the n i t r o g e n r e s u l t s p r e s e n t e d h e r e a l s o suggest t h a t s p i n - o r b i t e f f e c t s may be s i g n i f i c a n t f o r f i r s t row atoms bound to h e a v i e r elements. Copper p o r p h y r i n i s one of the b e s t - c h a r a c t e r i z e d of the m e t a l l o p o r p h y r i n s , and i t s e l e c t r o n s p i n resonance (ESR) spectrum has been known f o r a q u a r t e r of a c e n t u r y . ( 1 7 ) More r e c e n t l y , e l e c t r o n n u c l e a r double resonance (ENDOR) i n v e s t i g a t i o n s have p r o v i d e d the complete h y p e r f i n e t e n s o r s f o r the m e t a l , the n i t r o g e n s and the p y r r o l e p r o t o n s . ( 1 8 ) We have used t h i s d e t a i l e d knowledge earlier(_5,j3) to assess the q u a l i t y of s c a t t e r e d - w a v e c a l c u l a t i o n s . I n those c a l c u l a t i o n s , the c o n t r i b u t i o n s from e l e c t r o n i c o r b i t a l m o t i o n (induced by s p i n - o r b i t m i x i n g ) were e s t i m a t e d from c r y s t a l f i e l d t h e o r y ( f o r the copper atom) or were n e g l e c t e d ( f o r the n i t r o gen and hydrogen atoms). Here I d i s c u s s f o r the f i r s t time d i r e c t c a l c u l a t i o n s of these c o n t r i b u t i o n s to the copper and n i t r o g e n h y p e r f i n e t e n s o r s , as w e l l as to the m o l e c u l a r ^ - t e n s o r . The theory of i n c l u d i n g magnetic p e r t u r b a t i o n s has been discussed earlier.(11,14-16) I n D i r a c t h e o r y , e x t e r n a l f i e l d s appear through the o p e r a t o r 3


1

H

f

= e α·Α

(4)

where Of i s a g a i n a v e c t o r of D i r a c m a t r i c e s and A i s the v e c t o r p o t e n t i a l of the e x t e r n a l f i e l d . I n the case of a c o n s t a n t e x t e r n a l f i e l d Β (the Zeeman e f f e c t ) , A = (Β χ r ) / 2 . For the h y p e r f i n e term, A = (μ~χ r ) / r where μ i s the n u c l e a r magnetic moment. M a t r i x elements of these o p e r a t o r s are e v a l u a t e d i n the b a s i s spanning the two rows of the e3g r e p r e s e n t a t i o n of the h i g h e s t o c c u p i e d o r b i t a l . The a n g u l a r i n t e g r a l s r e q u i r e d may be performed a n a l y t i c a l l y , and the r a d i a l i n t e g r a l s are done n u m e r i c a l l y . The r e s u l t i n g e n e r g i e s a r e then f i t t o the s p i n H a m i l t o n i a n 3



64 Hspin = S-g-B

+ S·Α ·I η

(5)

n

where S = 1/2, I i s the nuclear spin operator, and η = Cu or N. Table II gives calculated results for copper porphine and experimental values for copper tetraphenylporphine. Results at the n o n - r e l a t i v i s t i c l i m i t (c = °°) were obtained from the same program by setting the speed of l i g h t to a very large number ( 1 0 a.u.). As we have discussed elsewhere,(14-16) numerical errors i n computing the matrix elements of the Zeeman operator are s i g n i f i c a n t , so that i n the n o n - r e l a t i v i s t i c l i m i t one does not obtain a g-tensor equal to 2. In other work, we have found i t to be a useful approximation to assume that these numerical errors cancel i n the n o n - r e l a t i v i s t i c and r e l a t i v i s t i c calculations, so that the computed difference may y i e l d a good estimate of Ag Ξ g - 2.0023. Results using this approx imation are given i n part A of Table I I , and show approximate, although not quantitative, agreement with experiment. For squareplanar copper complexes, equally good agreement i s generally possible using ligand f i e l d theory, so that the present results provide l i t t l e new insight. We expect, however, that i f this l e v e l of agreement with experiment carries over to more complicated cases (e.g. with lower symmetry or more than one metal atom,) the DSW model may pro vide useful q u a l i t a t i v e models for molecular Zeeman e f f e c t s , p a r t i cularly for cases with heavy atoms and large spin-orbit mixings. Hyperfine tensors are given i n parts Β and C of Table I I . Although only the t o t a l hyperfine interaction i s determined d i r e c t l y from the procedure outlined above, we have found i t useful to decom pose the t o t a l into parts i n the following approximate fashion: a Fermi term'is defined as the contribution from s-orbitals (which i s equivalent to the usual Fermi operator as c -> °°) ; a spin-dipolar contribution i s estimated as i n n o n - r e l a t i v i s t i c theory from the computed expectation value of 3 ( S * r ) ( I r ) / r ; and the remainder i s ascribed to the "spin-orbit" contribution, i . e . to that a r i s i n g from unquenched o r b i t a l angular momentum. On the basis of their ENDOR results and c r y s t a l f i e l d theory, Brown and Hoffman(18) have made empirical estimates of these three factors for the copper hyperfine interaction; these are shown i n Table I I . The spin dipolar contributions may be estimated from a , the population of the unpaired electron i n the Cu dxy o r b i t a l : n


15

e

5

2

A

d i p = -(4/7)Pa

2

;

A^P a n c

t

= (2/7)Pa

2

ie

(6)

where Ρ = ge3e8N^N > * * expectation value i s over the 3d. copper o r b i t a l . Both our e a r l i e r results(15) and the empirical analysis of Brown and Hoffman suggest that for the copper porphyrin, this value i s about 10% larger than the free ion value.(19) The proper value to use for a depends upon how overlap i s treated, but for the present q u a l i t a t i v e purposes i t i s s u f f i c i e n t to set i t equal to 0.62, the f r a c t i o n of unpaired spin on the metal estimated from both Χα and c r y s t a l f i e l d calculations.(5,18) The spin-orbit effect contributes both to the deviation of the molecular j*-tensor from the spin-only value, and to the metal hyper fine interaction. We can use experimental values of the former to estimate the l a t t e r : 2


Spin-Orbit and Spin-Polarization

CASE

4.

Table I I .

Magnetic Resonance Parameters for CuPor


c = A.

00

c

Zeeman interaction 1.761 II II 1.799 el C u hyperfine interaction A P -499 II dip 249 g

B.

6 3

di

A

A

so

0

II

0 so 1 *N hyperfine interactions A (Fermi) 52.2 A

C.

ll

Adip xx yy zz A

A

s o

t o t

65

Effects in Metalloporphyrins

diff.

0.155 0.059

1.915 1.858 (MHz) -492

exp.

0.188 0.043

a

D

-453

246

227

290

263

c

d

47

86 b

(MHz) 52.8

-3.7 7.4 -3.7

-3.9 7.6 -3.8

xx yy zz

-0.2 -0.1 -0.2

-3.7 1.1 -1.6

xx yy zz

48.2 59.5 48.2

45.2 61.5 47.5

42.8 54.2 44.1

e

^ a l u e of g (observed) -2.0023, from Reference 9. ^See text for method of calculation. E m p i r i c a l estimate from Equation (6) using Ρ = 1280 MHz and α = 0.62. E m p i r i c a l estimate using Equation (7), Ρ = 1280 MHz and the observed g-tensor. Source: Reproduced from Ref. 10· Copyright 1985 American Chemical Society. c

d

e


P O R P H Y R I N S : E X C I T E D STATES A N D

66 ASO = - Ρ [ ( 3 / 7 ) ( - ) + ( -


8θ

δ1

8θ

δ||

) ] ; k$£

= -P[ (6/7) ( g - g ) ] e

DYNAMICS

(7)

A

T a b l e I I compares these e m p i r i c a l e s t i m a t e s w i t h those o b t a i n e d from the DSW c a l c u l a t i o n . R e l a t i v i s t i c c o n t r i b u t i o n s have l i t t l e e f f e c t on the s p i n - d i p o l a r i n t e r a c t i o n s , and b o t h c a l c u l a t i o n s a r e i n r e a s o n a b l y good agreement w i t h the e m p i r i c a l e s t i m a t e s . The s p i n o r b i t c o n t r i b u t i o n s a r e a l s o i n moderately good agreement w i t h the e m p i r i c a l e s t i m a t e s , showing t h a t e l e c t r o n c u r r e n t s about the _za x i s a r e c o n s i d e r a b l y more i m p o r t a n t than those about axes i n the p l a n e of the l i g a n d . Indeed, i n v i e w of the a p p r o x i m a t i o n s t h a t e n t e r i n t o E q u a t i o n 7, (_5,8) , one might have as much c o n f i d e n c e i n the DSW r e s u l t as i n the e m p i r i c a l e s t i m a t e g i v e n i n the f i n a l column. S i n c e c o r e p o l a r i z a t i o n e f f e c t s a r e not i n c l u d e d i n the p r e s e n t DSW c a l c u l a t i o n s , no Fermi c o n t r i b u t i o n to the m e t a l hyper f i n e i n t e r a c t i o n a r i s e s from the p r e s e n t w a v e f u n c t i o n , a l t h o u g h i t s c o n t r i b u t i o n to the e x p e r i m e n t a l t e n s o r i s s i g n i f i c a n t . We have d i s c u s s e d such c o r e p o l a r i z a t i o n e f f e c t s elsewhere.(5,8) For the n i t r o g e n h y p e r f i n e t e n s o r s , t h e r e i s no s a t i s f a c t o r y e m p i r i c a l scheme f o r e s t i m a t i n g the v a r i o u s c o n t r i b u t i o n s , so t h a t T a b l e I I compares the t o t a l observed t e n s o r to the DSW r e s u l t . The t e n s o r s a r e g i v e n i n t h e i r p r i n c i p a l a x i s system, w i t h z^ p e r p e n d i c u l a r t o the p l a n e of the heme and y_ a l o n g the Cu-N bond. The s m a l l v a l u e s (0.1 - 0.2 MHz) found f o r A 0 i n the n o n r e l a t i v i s t i c l i m i t a r e not a consequence of o r b i t a l m o t i o n (which must v a n i s h i n t h i s l i m i t ) but a r e the r e s u l t of i n a c c u r a c i e s i n the d e c o m p o s i t i o n of the t o t a l t e n s o r i n t o i t s components, as d e s c r i b e d above. A l t h o u g h a l l of the c a l c u l a t e d v a l u e s are i n moderately good agreement w i t h experiment, of p a r t i c u l a r i n t e r e s t h e r e i s the d e v i a t i o n of t h i s i n t e r a c t i o n from b e i n g a x i a l about the Cu-N bond, i . e . t h a t Αχχ φ Αζζ· I n a s p i n - r e s t r i c t e d , n o n r e l a t i v i s t i c t h e o r y , t h i s t e n s o r w i l l be n e a r l y a x i a l — s e e , f o r example the c = results of T a b l e I I . Each of the two mechanisms c o n s i d e r e d i n t h i s p a p e r — s p i n p o l a r i z a t i o n and s p i n - o r b i t c o u p l i n g — m i g h t c o n t r i b u t e t o d e v i a t i o n s from a x i a l i t y . Indeed, n o n - r e l a t i v i s t i c c a l c u l a t i o n s i n c l u d i n g s p i n - p o l a r i z a t i o n e f f e c t s ( 8 ) i l l u s t r a t e d one p o s s i b l e mechanism: exchange e f f e c t s can a l t e r the s p i n p o p u l a t i o n s on the n i t r o g e n s such t h a t the p o p u l a t i o n i n the o u t - o f - p l a n e ρττ o r b i t a l i s g r e a t e r than t h a t f o r the i n - p l a n e ρπ o r b i t a l by about 0.01. The s p i n - d i p o l a r i n t e r a c t i o n w i l l then y i e l d v a l u e s of Azz - Αχχ of about 1.3 MHz, i n agreement w i t h experiment.(18) Furthermore, e x a m i n a t i o n of the n a t u r e of the m o l e c u l a r o r b i t a l s suggests t h a t the s p i n - o r b i t i n t e r a c t i o n s h o u l d be unable to i n d u c e ρπ p o p u l a t i o n s t h a t a r e t h i s l a r g e , (8,JL8) l e a d i n g t o the n o t i o n t h a t s p i n o r b i t e f f e c t s may be unimportant f o r a q u a l i t a t i v e u n d e r s t a n d i n g of the n i t r o g e n h y p e r f i n e t e n s o r s i n m e t a l l o p o r p h y r i n s . The r e s u l t s i n T a b l e I I suggest t h a t t h i s c o n c l u s i o n may be i n c o r r e c t : o r b i t a l m o t i o n of the e l e c t r o n s can c o n t r i b u t e t o the h y p e r f i n e t e n s o r amounts comparable t o the s p i n - d i p o l a r i n t e r a c t i o n . T h i s does not a r i s e from s i g n i f i c a n t changes i n s p i n p o p u l a t i o n s — the i n - p l a n e and o u t - o f - p l a n e ρπ p o p u l a t i o n s i n the p a r t i a l l y o c c u p i e d o r b i t a l a r e 1 χ 10~ and 4 χ 10 *, r e s p e c t i v e l y . As a r e s u l t , the v a l u e s f o r dip a r e n e a r l y the same as i n the non r e l a t i v i s t i c c a l c u l a t i o n . R a t h e r , a new mechanism i s a t work, i n s

00

6

-1

A



4.

CASE

67


w h i c h the magnetic f i e l d a s s o c i a t e d w i t h o r b i t a l m o t i o n i n t e r a c t s w i t h the magnetic d i p o l e moment of the n i t r o g e n n u c l e u s . Since t h i s i n t e r a c t i o n i s n e i t h e r i s o t r o p i c nor t r a c e l e s s , i t can g r e a t l y comp l i c a t e the a n a l y s i s of the observed r e s u l t s . I n p a r t i c u l a r , the a s s o c i a t i o n of the i s o t r o p i c and a n i s o t r o p i c p o r t i o n s of the observed t e n s o r w i t h F e r m i and s p i n - d i p o l a r mechanisms i s no l o n g e r a p p r o p r i a t e , and i t i s c l e a r l y r i s k y to attempt to d e r i v e bonding i n f o r m a t i o n from such an a n a l y s i s . U n f o r t u n a t e l y , the p r e s e n t r e s u l t s a r e a l s o l i m i t e d . I n a d d i t i o n to e r r o r s a r i s i n g from the exchange p o t e n t i a l and from the m u f f i n t i n a p p r o x i m a t i o n s , we have no means a t p r e s e n t t o i n c l u d e b o t h exchange p o l a r i z a t i o n and s p i n - o r b i t e f f e c t s i n t o t h e s e c a l c u l a t i o n s , a l t h o u g h each i s c l e a r l y of p o t e n t i a l importance i f a c c u r a c y g r e a t e r than 5 Mhz or so i s to be a c h i e v e d . A l l of the c a l c u l a t i o n s and e m p i r i c a l a n a l y s e s agree t h a t 7 - 9% of the u n p a i r e d s p i n i n CuPor r e s i d e s on each n i t r o g e n atom; more d e t a i l e d c o n c l u s i o n s w i l l p r o b a b l y have t o await the development of r e l i a b l e c o m p u t a t i o n a l schemes t h a t can i n c o r p o r a t e a l l of the s m a l l e f f e c t s t h a t c o n t r i b u t e to l i g a n d h y p e r f i n e i n t e r a c t i o n s . Exchange P o l a r i z a t i o n i n I r o n P o r p h i n e F i g u r e 1 shows the o n e - e l e c t r o n e n e r g i e s of i r o n ( I I ) p o r p h i n e i n b o t h s p i n - r e s t r i c t e d and s p i n - u n r e s t r i c t e d ( d i f f e r e n t o r b i t a l s f o r d i f f e r e n t s p i n s ) form. The geometry used has a p l a n a r p o r p h i n e w i t h an Fe-N d i s t a n c e of 2.01 Â and no a x i a l l i g a n d s . T h i s c a l c u l a t i o n assumes a h i g h - s p i n q u i n t e t s t a t e w i t h the c o n f i g u r a t i o n ( d x y ) (dxz) (dyz) (dz ) (dx -y )\ giving B symmetry. The d i f f e r e n c e s between the l e f t - and r i g h t - h a n d - s i d e s of t h i s diagram a r e s t r i k i n g and important i n i n t e r p r e t i n g the p r o p e r t i e s of t h i s and o t h e r h i g h s p i n i r o n complexes. S i n c e the s p i n d e n s i t y i s p r e d o m i n a n t l y l o c a t e d on the i r o n atom, exchange f o r c e s s p l i t the i r o n d. l e v e l s by about 3.5 eV, r e f l e c t i n g the f a v o r a b l e exchange i n t e r a c t i o n s among the m a j o r i t y s p i n (a) e l e c t r o n s compared to those of s p i n 3. (There i s l i t t l e s p i n d e n s i t y on the l i g a n d , so t h a t the α and 3 e n e r g i e s are n e a r l y the same; i n the f i g u r e , o n l y one l i n e i s used to r e p r e s e n t b o t h e n e r g i e s . ) The α o r b i t a l s (except f o r d x - y ) are low i n energy and are always o c c u p i e d i n the ground and lowl y i n g e x c i t e d s t a t e s . The o t h e r 3 s p i n cl o r b i t a l s are near the Fermi l e v e l , and are i m p o r t a n t i n d e t e r m i n i n g s p e c t r a and r e a c t i v i t i e s . I t i s these o r b i t a l s , r a t h e r than the average of α and 3 o r b i t a l s seen i n s p i n - r e s t r i c t e d t h e o r i e s , whose c h a r a c t e r s and e n e r g i e s a r e important f o r q u a l i t a t i v e c o n s i d e r a t i o n s . As an example, c o n s i d e r the e x c i t e d s t a t e c r e a t e d by a p r o m o t i o n from the i r o n dxy o r b i t a l i n t o one of the unoccupied o r b i t a l s on the p o r p h y r i n l i g a n d . I n a s p i n - r e s t r i c t e d theory (such as extended H u c k e l theory or the l e f t - h a n d s i d e of F i g u r e 1) the e x c i t a t i o n energy i s not approximated by the c o r r e s p o n d i n g d i f f e r e n c e o r b i t a l e n e r g i e s ; r a t h e r one must c o r r e c t t h i s w i t h an exchange term. I f a 3 s p i n e l e c t r o n i s promoted, the exchange c o r r e c t i o n i s about -1.2 e V , ( 2 0 - 2 2 ) i . e . , the f a v o r a b l e exchange i n t e r a c t i o n s i n the r e s u l t i n g h i g h - s p i n F e ( I I I ) s p e c i e s make the e x c i t a t i o n energy s m a l l e r than the o n e - e l e c t r o n energy d i f f e r e n c e . On the o t h e r hand, i f an α e l e c t r o n i s promoted, the c o r r e c t i o n i s +1.8 eV, a r e f l e c t i o n of the l o s s of 2

1

1

1

2

2

5

2 g

2


2

68


exchange correlation i n the r e s u l t i n g f e r r i c state. The right hand side of Figure 1, however, c l e a r l y shows that the promotion of an α dxy electron i s more costly than promotion of a 3 one by about the 3 eV difference i n the two "correction" factors cited above—to a large extent the exchange corrections are already included i n the one-electron energies. Table I I I shows the populations f o r the " c r y s t a l f i e l d " o r b i t a l s , i . e . f o r those occupied o r b i t a l s with greater than 10% popula tion on the i r o n atom. The comparison i n this table i s between the populations f o r the B state (upper l i n e i n the table) and a A g state (lower l i n e ) with occupation (dxy) (dz ) (άπ) , which i s l i k e l y to be a major component of the ground state. (9) The triplet-* quintet change thus corresponds to the o r b i t a l promotion d z 3 + dx -y q. The results i l l u s t r a t e the ways i n which spin-exchange affects the electronic structure of this complex. The energies of the 3-spin o r b i t a l s are nearly unchanged but the strongly l o c a l i z e d α o r b i t a l s (6b2g0t, 8ai ot and 7bi a) are lower i n energy i n the quintet state by about 1.5 eV due to more favorable exchange i n t e r actions i n the high-spin state. The s i t u a t i o n i s more complex f o r the eg o r b i t a l s , which can mix strongly with prophyrin π o r b i t a l s of the same symmetry. In the quintet state, the άπα o r b i t a l s are s t a b i l i z e d by this i n t e r a c t i o n , and are the lowest of a l l the dl i k e l e v e l s , appearing mainly i n the 2ega l e v e l , with about 9% character i n the 4e a o r b i t a l . In the t r i p l e t , both 2e and 4e have s i g n i f i c a n t dTT population, a feature that makes i t d i f f i c u l t to apply c r y s t a l f i e l d theory to this complex.(9) We have discussed the o p t i c a l absorption spectrum of t r i p l e t FePor i n our e a r l i e r paper.(9) Similar results f o r the quintet state, p a r t i c u l a r l y f o r charge transfer t r a n s i t i o n s , are of interest as possible explanations f o r absorption bands seen the near-infrared region of deoxy heme proteins.(20-22) Both metal-to-ligand and ligand-to-metal transitions need to be considered. A t y p i c a l ligandto-metal promotion would be from the highest occupied porphyrin a2uM o r b i t a l into 4 e 3 , which i s 76% d r r ; a c h a r a c t e r i s t i c metalto-ligand t r a n s i t i o n i s from dxy3 to the lowest unoccupied porphyrin e (7T) o r b i t a l . Since the energies of both the porphyrin o r b i t a l s and the metal spin-3 o r b i t a l s are nearly the same i n the t r i p l e t and quintet states, the excitation energies should also be about the same. For the t r i p l e t state, we have shown that Χα calculations predict both metal-to-ligand and ligand-to-metal transitions to l i e i n the 1.5 to 2.0 eV range,(9) and the same general conclusions should hold f o r high-spin states. We s h a l l report elsewhere(23) results f o r five-coordinate high spin complexes that support this conclusion. The calculations of charge transfer excitation energies, however, are s u f f i c i e n t l y uncertain to make i t hazardous to attempt detailed assignments of observed o p t i c a l spectra. Unlike the s i t u a t i o n f o r charge transfer energies, d-d. excitations are expected to be quite sensitive to spin state ( t r i p l e t vs. quintet) and the presence of a x i a l ligands. Calculations of d-d transitions for high-spin five-coordinate iron porphyrins w i l l be presented elsewhere. 5

3

2 g

2


2

2

2

2

2

2

2

g

g

g

g

g

g


g

CASE

4.


FePor

dxV

5

B

2Q

N

dxz'dyz' dxy**

-7 ''V

J n

Qiu °2l 2- 2a

x

dx


69

J

y

°2u -eg

-b

1u

dz -dxy "dxz ,dyz 2 a

a

e

Restricted

Figure 1. Table I I I . Orbital

a

Unrestricted

O r b i t a l energies f o r i r o n porphine.

Crystal F i e l d Orbitals f o r FePor

3

Populations

Energy, eV

6b e

- 7.18 - 7.31

.973 Fe dxy .970 Fe dxy

7b

- 7.82 - 6.31

.665 Fe d x - y .744 Fe d x - y

- 8.86 - 8.45

.088 Fe .382 Fe

dTT

- 9.24 - 9.28

.112 Fe .110 Fe

dTT

-10.67 - 9.15

.105 Fe s + .845 Fe d z .107 Fe s + .871 Fe d z

6b cx

-10.89 - 9.26

.950 Fe dxy .965 Fe dxy

2eg a

-11.07 - 9.93

.804 Fe dTT + .009 Ca + .011 C3 + .009Cm .557 Fe dTT + .033 Ν + .021 C3 + .020 Cm

2 g

l g

«

4eg a 3eg Β

8

a

i g

a

2g

drr

dTT

2

2

z

z

.049 Ν go + .018 Ns .036 Ν ρσ + 0.16 Ns

+ +

+ .090 Ν + + .062 Ν +

.061 C3 .037 C3

+ .066 Ν + + .049 Ν +

.078 C3 .083 C3 2

2

Populations per atom. A l l o r b i t a l s shown are occupied. For each o r b i t a l , the upper l i n e gives results f o r the B state, and the lower l i n e f o r the A g state. 5

2 g

3

2


P O R P H Y R I N S : E X C I T E D STATES A N D

70

DYNAMICS


Conclusions I n t h i s paper I have t r i e d t o o u t l i n e some o f the q u a l i t a t i v e f e a t u r e s o f s p i n - o r b i t m i x i n g and exchange p o l a r i z a t i o n e f f e c t s i n metalloporphyrins. A l t h o u g h the c o m p u t a t i o n a l models i n v o l v e d a r e crude ones, the g e n e r a l f e a t u r e s found h e r e a r e expected t o p e r s i s t i n more a c c u r a t e c a l c u l a t i o n s . One improvement t h a t would c l e a r l y be d e s i r e a b l e i s t o be a b l e t o i n c o r p o r a t e b o t h e f f e c t s i n a s i n g l e model. I t i s not easy t o see how spin-exchange e f f e c t s can be i n c o r p o r a t e d i n t o the D i r a c model, s i n c e the o r b i t a l s and s t a t e s are s p i n m i x t u r e s . I n p r i n c i p l e , b o t h e f f e c t s can be i n c l u d e d through p e r t u r b a t i o n t h e o r y o r by c o n f i g u r a t i o n i n t e r a c t i o n , ( 2 4 , 2 5 ) but i t remains to be seen i f s i m p l e p h y s i c a l p i c t u r e s can be d e r i v e d from these c a l c u l a t i o n s . Other approximate methods o f i n c l u d i n g b o t h e f f e c t s have been t r i e d , ( 2 6 , 2 7 ) and i t i s t o be hoped t h a t u s e f u l models a l o n g these l i n e s can be developed i n the next few y e a r s . There i s a l s o a g r e a t need f o r c a l c u l a t i o n s o f g r e a t e r i n t r i n s i c r e l i a b i l i t y than those r e p o r t e d h e r e . I n b o t h o f the s p e c i f i c i n s t a n c e s d i s c u s s e d above ( h y p e r f i n e t e n s o r s i n CuPor and o p t i c a l e x c i t a t i o n e n e r g i e s i n FePor) the g e n e r a l f e a t u r e s o f the Χα r e s u l t s s h o u l d be r e l i a b l e but the e r r o r s are s u f f i c i e n t l y l a r g e t h a t d e t a i l e d i n t e r p r e t a t i o n s o f the experiments are o f dubious v a l i d i t y . Ab i n i t i o c a l c u l a t i o n s have g r e a t p o t e n t i a l , but s u f f e r from the l a r g e number o f b a s i s f u n c t i o n s r e q u i r e d , and from the f a c t t h a t the H a r t r e e - F o c k model i s not a v e r y good one f o r opens h e l l t r a n s i t i o n m e t a l c o m p l e x e s — f o r example, i t o v e r e s t i m a t e s by a l a r g e amount the s t a b i l i t y o f h i g h - s p i n complexes r e l a t i v e t o i n t e r m e d i a t e o r l o w - s p i n c o n f i g u r a t i o n s . U n t i l methods o f i n c l u d i n g and u n d e r s t a n d i n g c o r r e l a t i o n e f f e c t s i n these systems are d e v e l oped, the c o n n e c t i o n between theory and experiment w i l l c o n t i n u e to be e x c i t i n g and i l l u m i n a t i n g , but not d e f i n i t i v e . T h i s work was s u p p o r t e d i n p a r t by a g r a n t from the N a t i o n a l Science Foundation.

Literature Cited 1. Zerner, M.; Gouterman, M. Theor. Chim. Acta 1966, 4, 44. See also Gouterman, M. in The Porphyrins, Vol. III. Physical Chemistry, D. Dolphin (ed.) Academic Press, 1977. 2. For reviews see: Loew, G.H., in Iron Porphyrins, Part I, A.B.P. Lever and H.B. Gray, eds. Addison-Wesley, 1983, p. 1. Dedieu, Α.; Rohmer, M.-M.; Veillard, A. Adv. Quantum Chem. 1982, 16, 43. 3. Johnson, K.H. Adv. Quantum Chem., 1973, 7, 143. 4. Case, D.A. Annu. Rev. Phys. Chem. 1982, 33, 151. 5. Case, D.A.; Karplus, M. J. Am. Chem. Soc. 1977, 99, 6182. 6. Huynh, B.H.; Case, D.A.; Karplus, M. J. Am. Chem. Soc., 1977, 99, 6103. 7. Case, D.A.; Huynh, B.H.; Karplus, M. J. Am. Chem. Soc., 1979, 101, 4433. 8. Sontum, S.F.; Case, D.A. J. Phys. Chem. 1982, 86, 1596. 9. Sontum, S.F.; Case, D.A.; Karplus, M. J. Chem. Phys. 1983, 79, 2881. 10. Sontum, S.F.; Case, D.A. J. Am. Chem. Soc., 1985, 107, 4013.



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11. For a review, see Yang, C.Y.; Case, D.A. in Local Density Approximations in Quantum Chemistry and Solid State Physics, J.P. Dahl and J. Avery, eds. Plenum, 1984, p. 643. 12. Lopez, J.P.; Yang, C.Y.; Case, D.A. Chem. Phys. Lett. 1982, 91, 353. 13. Atherton, N.M. Electron Spin Resonance, Theory and Applications. (John Wiley, 1973), Section 6.4. 14. Arratia-Perez, R.; Case, D.A. J. Chem. Phys. 1983, 79, 4939. 15. Case, D.A.; Lopez, J.P. J. Chem. Phys. 1984, 80, 3270. 16. Case, D.A. J. Chem. Phys. 1985, 83, 5792. 17. Manoharan, P.T.; Rogers, M.T. Electron Spin Resonance of Metal Complexes, Yeh, T.F., Ed. Adam Hilger: New York, 1969, p. 143, and references therein. 18. Brown, T.G.; Hoffman, B.M. Mol. Phys. 1980, 39, 1073. 19. McGarvey, B.R. J. Phys. Chem. 1967, 71, 51. 20. Eaton, W.A.; Hanson, L.K.; Stephens, P.J.; Sutherland, J.C.; Dunn, J.B.R. J. Am. Chem. Soc. 1978, 100, 4991. 21. Zerner, M. Ph.D. thesis, Harvard University, 1966. 22. Makinen, M.W.; Churg, A.K. Iron Porphyrins, Part I, Lever, A.B.P. and Gray, H.B., eds. (Addison-Wesley, 1983), p. 141. 23. Sontum, S.F.; Case, D.A.; Karplus, M. manuscript in preparation. 24. See, e.g. Mishra, K.C.; Mishra, S.K.; Das, T.P. J. Am. Chem. Soc. 1983, 105, 7729. 25. Vajed-Samii, M.; Andriessen, J.; Das, B.P.; Ray, S.N.; Lee, T.; Das, T.P- J. Phys. Β 1982, 15, 1379. 26. Weinert, M.; Freeman, A.J. Phys. Rev. Β 1983, 28, 6262. 27. Ellis, D.E.; Goodman, G.L. Int. J. Quantum Chem. 1984, 25, 185. RECEIVED March 28, 1986


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