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Relativistic molecular orbital theory of zero-field splittings in triplets
\‘ohime
109,
CHE%llCAL
nun~ber J
RELATIVISTIC
~~OLECULAR
ORBITAL
PHYSICS
THEORY
LETTERS
OF ZERO-FIELD
3 August
SPLITTINGS
1984
IN TRIPLETS
David A. CASE L)Pparlrrrivrt
of
C~~ctrlislry.
l.hiwrSiQ*
of Caii/bnIia, DalTs. California 95616, UC?.4
:I n~cthod of valculating mnrris eien~enrs of I/r,, in B basis of IIirw scattered-wave (DSW) orbitals is outlined. In the IimiI c - cu. Ihis nwthod rcduccs to 11x11described by Cook and KarpJus for non-relativistic orbitals. For triplet states that can bc degribcd by a single confiiurntiun with two unJxdred electrons, rhe relativistic exchange intgralsgive nor only the sin_&ct --triptct splitiings (as in non-reiativistic tf~eory), but also tfre spin-orbJt contributions to rfte triplet zero-field splittings. wl’hicf~ has a wry large vahc of D fte5tIlls arc reprrrred for the 3111 * n*) excited state of ~-tliiop~~ro~c (4J~-pvr~-rl-thionc). tc;i1c. -31 WI-1, e?rp_ -23 to -28 cm-1).
1. Introduction
2. Method of calculation
Iilcctror~ic strlict ure calculations that use an effective cr;charlgc-~orrcl;tioll potcnti31 (such 3s tfje Xa or local spin density functions) can bc viewed in two difI’crc111 lights. From OIlC point of view (recc11tly revicwcd by Parr [ I]), the Xff method is an approximarion to csact density functional theory, and improvc~mnts way he possible as we gain a better understnnding of deusit y l’unct ional theory of molecular ground states. A second point of view(r~iucli closer to that hcld by Slatcr 121) views the Xa method as an zpproximate molecular orbital theory, applicable to both gr~mnd aacf excited smcs, aud capable of being itnproved by the staudard methods of quantum chemistry. e.g. through configuration interaction. In this light, Ihc Xo ii~cthod can bc vicwcd as a convcnicnt Incaus nf’gcricraliug ;Lbusis scf. for use in further calculations itI which 00 approxinmtiom are made to the f~aniiltortiatt. This was the approach adopted in early cnlculatious by Zarc [3] oi‘atomie nndtiplet splittings. and mm recently by Cook ard Karplus [4,5]. in scattcrcdwave calc~~latioris of siuglct-triplet splittings in ozone. I lerc I show how thcsc methods can be cxtcnded to the lclstivistic Dim scattcrcd-wave (DSW) urethod, and as an cXantl)Ic. report calcl)lations of the zero-field splittings in the 11-+ zr* excited state of a tltiokctonc.
The DSW method uses the smc potential approximations as non-relativistic scattered wave theory, but starts from the Dirac rather than the Schrijdinger equation. its formulation and applications have recently been reviewed [6--81. For each orbital, the wavefunction inside the atomic spheres is written as a linear combination of spinors, of the form.
60
(1) where C(&; Xktt1 = c s=+ l/2
111- s. s) Y;,,--,(0,
$?J)X(s) .
(2)
Here g(r) and fir)arc radial functions, determined by numerical integration of the radial Dirac equations 191. Y is a spherical harmonic, C is a Clebsch-Gordan coefficient. and x(s) a spii~ function, x(i) = (A), x(-i> = ((1). These spinors are cigenfunctions of J” (with cigcuvalue j(j + I)?$, where j = Ik I -+) md Jz (with cigcnvahc m?~). The orbital energies are determined by the DSW secular equations to yiefd a cont~uous wnvefunction, and the resulting orbit& transform 0 009-26 14/84/S 03 .OO 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
Volume 109, number 1
CHEhlICAL PHYSICS LETTERS
according to the “extra” irreducible representations of the molecular double point group. r As in non-relativistic theory, the wavefunction in the intersphere region is expressed 3s a multi-center combination of free-electron functions. Following earlier work [4,5,10], we approximate this intersphere part by expanding.the atomic r3dial functions beyond the sphere boundaries. This expansion accounts for all the intersphere charge, and partitions it among the atoms according to the surface charge densities in the atomic spheres. This approximation has been shown to be no worse than multiple scattering approximations themselves [ 1 I], and leads to results for molecular properties that are intermediate between minimum basis and extended basis ab initio Hartree-Fock calculations [6,1 1,121. Matrix elements of 1/rt2 can be evaluated if the operator is espanded in a convenient coordinate system. For single-center integrals, where all the wavefunctions belong to a single atom, this is the ordinary Inplace-type expansion.
The radial integrals may be performed numerically, and the angular integrals (knll YIIk’nz’) are given, to within a normalization constant, by the & coefficients of Grant [ 13,141. This procedure has been used for many years in atomic Dirac calculations, and complete details are given elsewhere [ 141. Note that the main difference from the corresponding non-relativistic result lies in the replacement of & “Gaunt” coefficients of Condon and Shortley [15] by the analogous & coefficients defined over spinors. Minor differences inc!ude replacement of Dirac for Schr6dinger radial functions, and the use of complex rather than real spherical harmonics. For two-center integrals, Cook [S] has shown how the bipolar 1/‘t2 expansions developed by Steinborn and Filter [lG] can be adapted to the multiple sc;lttering problem. Two situations arise, one where each center is an atomic sphere, and one connecting an atomic sphc~e to the outer sphere that surrounds the molecule. Use of the complex form of these expansions, along with the relativistic Gaunt coefficients, dir&It yields an algorithm for the relativistic twocenter integrals. Since the scattered-wave orbitals obey a zero differential overlap criterion, three- and
3 Augst
1984
four-center integrals do not appear. Details of the computational scheme (including the way in which various terms are stored and retrieved) are available from the author. In the limit c + 00, this scheme reduces to that described by Cook and Karplus [4] for non-relativistic calculations. 1 have reproduced their results for ozone, using a value of lOI au for c. The accuracy of these results, compared to ab initio Coulomb and eschange integrals 141, suggests that they should be useful for many purposes.
3. Zero-field
splittings
An interesting and instructive 3pplintion of relativistic configuration interaction consists of estimating the spin-orbit contribution to zero-field splittings in triplets. Consider the case in which t!ie triplet can be approximated by a (Dirac) h.10 description with two unpaired electrons. Except in very high or very low symmetry, each orbital will transform according to a 2 X 2 irreducible representation of the double molecular point group [ 171. (For the C2, case considered here, ail the Dirac orbitals belong to y5_) We cgn ad3pt a convenient non-relativistic notation to this case. using an unbarred symbol to represent an orbit31 corresponding to row 1. and a barred symbol to correspond to row 2. Hence one can construct four determinants with one electron each in orbitals a and b: Ai = labj. A, = lii61. A3 = 1361, and A3 = Ilbl. In the absence of electron repulsion, these are degenerate, but they will mix and split by the action of l/r,?. It is easy to show that A, and A2 span the B, + B, representations of Czv, whereas A3 and A3 span A, + A?. Hence the 4 X 4 CI mat*ix splits into two 2 X 2 matrices: (4) where Q = (aElb6) fl=J=g
-Jab
- (36lb5), -K,,-+k-,
.
y = (a51 6b) - (ab 1%) _
I use a standard notlltion [ 181 for the two-electron integrals. such that f.& s (aa Ibb) and K3b s (sb I ba), 67
Vllllrnlc
~llld
Illilnll~r
I 09.
;I c0111111011
cli:~gw~l
cic~llc~lts
e = +a. p 2
y
CIIINICAL I’IIYSICS LI:TTl:.IIS
I
cleIllcllt ofeq.
has
bCCl1
sublractcd
l’rolll
(4). The cigcnvnhIcs
1llC
.
-4b
(6)
(7)
lo give f) = Iyl -- (3, Ii:‘1 = ck. Ilcncc cvalualion eight rwo-clcctru~l intcgwls of cq. (5) directly
of tlic
to a thioketone the /.cro-field splittings
not by spirl-orbit
coupling,
but rather
by spin-spin dipolar in1cracrions. which are no1 includcd in the theory prrscutcd above. RccerItly it has been found that ccrrain II - 7r* triplet sIaIcs, cspecially those conIairiiIig a sccorid row atoni. can have valucs of I> greater tluui 10 cIIi~- I. illi order of magnitude larger than typical spin-spin interactions. One ofthe simplesr of Ihcse is 41 I-pyran-4-thione (PT. fig. I ), for \vliicll II vulues of -24 and -28 cu- I have been obw vcd for two distiiw trap sides [20]. X conventional non-relativistic Xck multiple scatIcriug s:~lcula~ion ]2 I ] on I’T gives t hc rcsults shown iu fig. I. TIIc geonicIry was that found in a recent niicI-owave study ]22]. and sphere radii were chosen by Illc Ko1ni311 procedure 123 J with a scale factor of 02%. Partial waves through I = 2 were included for the ouler sphere and S, Ihrough I = I for C and 0. and I = 0 for I I. The two highest occupied orbitals are Gb, (priniarily loue pair, with 33% of the charge associated with the p_,.orbital on sulfur) and 3bI (a ITorbital with about 702 occupation on S). The lowest unoccupied orbital
is 3b t (E* with about
Upon going to a Dirac calculation, 68
PT
2 z -6E El
_6b2 3”, -lo
2
yields
Ilic /em-field spli11iIigs. II is worIlIwhilc to cwnparc this to the corrcspondiug mm-relativistic calculation. There a and ii have Ihc sa111cOIbiral part aud differ only iu their spin coulpo~wuts. In tlIis case it is easy to set that o = 0 and 0 = I-yl. so tht ccl. (6) yields three degcncrate triplet suhlcvcls and a siuglct separated from thcni by ZK,,,. Tlw ~ffcc~ of I IIC spin-orbit tern1 implici1 in the Dirac I IauIiltoui:III is 10 rcuiove this triplet state dcgcncracy.
For many organic triplets,
-4-
l
WHY,&= f>(SJ -- $.s’, + /fc.s~; -- s>:,
31-c don~htc’d
b
arc
III Ihe prcsc111 case (see below), lol Q /3 = lrl, and the three lowest eigcnvalucs cau bc fit to a conventional spin I13111i11011i311 [i’,j
4. Application
3 August 1984
10% S character).
there is significant
-8 -2b
t
spin-orbit mixing betwccu 6b2 and 3b,, yielding a top occupied orbital which is primarily sulfur lone pair, with sonic n cltaractcr ntixcd in. Shnilar behavior is seen if the non-relativistic calculation is converged with a single electron in bb, and 4bI (corrrcsponding 10 a misturc of I(n -+ n*) and 3(n + n*) sIaIcs). It is the potential from this latter calcuhuion that was used to obtain the results reported below. Alrl~ougl~ Dirnc spinors have four components, for the purpose of qualitative analysis we can ignore the bottom two. “snIall” conlponeuts, and identify the top two components with spin Q and /3. If WC further assun~ that the radial functions of thcj = I + i and j = I - i spinors arc the same (an excellent approxiniaIion for light atoms) we can decompose the Dirac orbitals into a I’auli form consistiug of (real) spherical IIarnIonics Inultiplied by spin functions. The results of this deconiposition are shown in table 1 for “row 1” of the irreducible representation ys. (Row 1 is defined to transforIii in the SdlllC way as a spin function with j =- i, m = -4 [ l7]-) Since both the direct product (b 1 X CC)and (b, X a) belong to row I of rs, the Dirac orbitals are nearly pure Q’spin states; small admixtures of a1 and a2 orbitals provide some 0 occupation_ The real part of the a function (component 1) corresponds to the lone pair (b?) orbital, and the imaginary part belongs to the b, (nj orbital. As table 1 indicates, the lower partially occupied orbital (which 1 call “a”) is mostly (91%) sulfur lone pair with some a character. (The ratio of lone pair to a coefficients on sulfur is
Volunic 109. number 1
3
CIiEhlICAL l’I1YSICS LETTERS
Aug11sr
1984
Table 1 Orbital characters Pauli decomposition -
Orbital a(“6b,
“) =
b(“4b;‘)
=
0.9518 -0.0595 +0.1363 -0.0027i + SIWlhx
s s Cy S
+ + -
0.0173i 0.05683 0.0165i 0.003Si
S S Cy S
I’.\ d,=
Q Q
1,:;.11~
a Q n D
!‘S d,2
;
I’s d AZ I’s I’.\I’s
a 0 Q * (r
+
O.OOlSi O.OOOli 0.0001 i 0.0003i 0.0002i
S S Cy Ca 0
i’_J $*z I’?’ I’_V P.1’
Q Q Q Q 0
p,. d.-
terms
-0.2090 s +0.3012 s -0.6282 Cy +0.4100 Ccu -0.3339 0 + s111a1kr tcnns
_-
55 : 1). The higher partially occupied orbital (“b”) is of 77character with a very small amount of in-plane character (a coefficient ratio of 150 : 1 for the sulfur p-orbitals). (Interestingly, the dr occupation on sulfur in orbital b is even larger than the prr occupation; I plan a future study on the effects of sulfur d-orbit& on thioketone zero-field splittings.) The relevant two-electron integrals arc given in table 2. The value of D is determined primarily by the difference between Kab and (ab Ii%), whereas E is detcrmincd niainly by (a6lbT). An esmination of
Table 2 Two-electron
integrals a)
( 1) non-relativistic b) J,,*=51194 J nn* = 45731
%rrl * = 9544 Knrr* = 631
(2) relativistic Jab = J& = 44273 c) (a:lbG)= -2.5 X 1O-7 (alllgb) = 2.3 X 10m5 (3) zero-field splittings d) D talc. = -31.5 I./I?!talc. = 0.26
&,t, = 428.3 (abli%)= -396.8 (a61 b?i) = 0.26
tsp. = -28. -14 esp. = 0.060. 0.063
a) All values in cm-‘. b) Dcsigations x, IT* and n rever to 3bt. 4bt, and 6b2, respectively. c) The equality of Jab and Jag is a consequence of symmetry in Czv; these integrals may not be equal in other point groups, such as C3v or Dgh. d) Esperimental values from ref. [ 101.
+ -
t lie C,,
COUpliilg coefficients [ 171 shows that, for the non-relativistic orbitals, K,,* = (mr* Iii*%). whereas A’,,,,+ and (nn*lii*ri) are equal inmagnitude. but have opposite signs. Hence a mixture of II and in clraracter in orbital “a” gives a &, not equal to (ab It%). Since the mr* exchange integral is an order of magnitude larger than that for nn* (see table 2), even a small admhrure of 71character into a lone pair orbital can yield significant zero-field splittings. It is not unreasonable to expect that other thioketones may exisr with even hrger values of D. and the observed sensitivity of D to the trap site is easy to understand in terms of very small changes in the amount of n-r mixing. Our value of E is larger than the experimental one by about a factor of four. This apparently arises from an overestimate of mixing to the Ia? (sr) orbital, which destroys the spin symmetry that otllerwise would make (a6lbZ) zero. In spite of this error, the theory does predict a value of E two orders of magnitude smaller than D, in accord with experiment. A similar calculation on the 3(5r + x*) state (with configuration 3b 1 + 4b 1) gives D = -0.85 cm- I. Although this value is not known experimentally for PT, it is in rough accord with observations on other thioketone n--71* states, such a thiouracil [24]. Thus the simple single configuration theory presented here appears capable of reproducing the large difference seen between 3(x +- .rr*) and 3(n + 7r*) states.
5. Discussion One principal
advantage
of the Dirac Hamiltonian 69
CHEMICAL PHYSICS LETTERS
Volume 109, num bcr 1
is that the spin-orbit interaction is contained in the zero-order states, and does not need to be added as a perturbation. This reduces the order of perturbation theory required to calculate properties that involve ;~I~quer~cl~ed orbital anguIar ~~~o~~~entu~~~. The usual theory [ 191 of the spin-orbit contribution to zerofield splittings involves a second-order perturbation stun over all cscited states. This can be very difficult to carry out since reliable infomlation about the nat urc and locarion of molecular excited states is rare& available. Such sun~s are not required in the theory presented here. Earlier [ZS] WC have used a similar approach t0 develop first (tather than second) order perturbation theories of orbital contributions to IIIOlecular hyperfine interactions. The calculations presented here can be extended in 1tv0 ways. First, spin-spin interactions can be included vh the Brcit operator. Expressions for the required one-center matris elements (which should be the nlost intlx)rtanl ones) have been dcvclopcd for use in atoniic calculations [ 141, and could easily be incorporated here. Second, the triplet state wsvefunction could be improved through the addition of nlore configurations. This would require the calculation of niatrix elements ot’one-particle 3s well as two-particle interactions. As with the 1/rll operator, this c;fn be accon~plished by a rclativcly straightforward nwdificstion of cxisring non-relativistic scattsred wave codes. This opens the
w:*y to 3 gencr31 Ci code for DSW wavefunctions: p10gran1s to carry this out ltre currently
under
de-
Vclopnlent. I:vcn wit/lout thcscextcnsions. however. the present results suscst that the sin~p!c theory here should bc uscii11 as a qualitative guide to zero-field splittings in
mhxules
with large spin-orbit
internetions.
Since
inan?; ursanic
arid transition mct;tl complexes I>11 in%0 llG.5 category. a large number of interesting spectroscopic q~plications can be cnvisioncd.
Acknowledgement
I Ihmk A.1 1. hlaki for pointing 0uL tfic interesting bcltsvior of PT. ;tnd ht. Cook for providing n copy of his non-relativistic codes. which were rhc basis for those described here. This work was supported by the National 3n Alfred
70
Science Foundation, I’. Slosn
Foundalion
CHE-!31-17101. Felbw.
DAC is
3 August 1984
References [l] R.C. Parr, Ann. Rev. Phys. Chem. 34 (1983) 631. [ 21 J.C. Slater. The self-consistent field for molecules and solids (hicCrawHil1, New York, 1974). R.N. Zarc, J. Chem. Phys. 45 (1966) 1966. M. Cook and M. liarplus, Ctlem. Letters 84 (1981) 565. ill. Cook, Ph.D. thesis, Harvard University (1981). D.A. Case, Ann. Rev. Phys. Chem. 33 (1982) 151. C.Y. Yang, in: Relativistic effects in atoms, molecules and solids, cd, G. hlalli (Plenum Press, New York, 1983) p. 335. (S] C.Y. Yang and D.A. Case, in: Local density approsimation: in quantum chemistry and solid-state physics, eds. J, Dahl and J.P. Avery (Plenum Press. New York, 1984) p. 643. [ 91 J.E. Ilarriman. Theoretical foundations of electron spin resonance (Academic Press. New York. 1978). [ 10 ] D.A. Case and 51. Karplus, Cbem. Phys. Letters 39 (1976) 33. [ 11 f $1. Cools and Xi. Karptus, J. Chem. Phps. 72 (1980) 7. [ ‘It ] D.A. C;tse, hf. Cook and M. Karplus, J. Chcm. fhys. 73 (1980) 3294. [ 13 1 I.P. Grant, Proc. Roy. Sot. A262 (196 1) 555. [ 141 I.P. Grant, Advan Phys. 19 (1970) 747. 1IS] E.U. Condon and G.11. Shortley, The theory of atomic spectra (Cambridge Univ. Press. London, 1935) ch. 6. [ 16 J E.O. Steinborn and E. Filter. intern. J. Quantum Chem. s9 (197.5) 135. 1171 G.F. Kostcr, J.O. Dimmock, R.G. \\‘bceler and 11. Statz, Properties of the 32 point groups (MIT Press, Cambridge. 1963). R.G. Parr, Quantum theory of molecular electronic structure (Benjamin, New York, 1964) section 5. N.%f_Atherton, Electron spin resonance. theory and applications (Wiley, New York, 1973) ch. 5. h1.R. Taherian and All. Maki. Chcm. Phys. Letters 96 (1983) 541:Chcnt. Phys. 68 (1982) 179: D.11. Uurlnnd. J. Chem. 1’11~s.75 (1981) 2635. Program No. 465 from Quantum Chemistry Program Exchange, Bloomington, Indiana. J.N. MacDonald, %A. %facKay, J.K. Tyicr, A.?. Cos and 1.C. ilwart. J. Chem. Sot. Faraday Ii 77 f 1981) 79. J.C. Norman Jr., Mol. Phys. 31 (1976) 1191. M.R. Taherian arId AM. Maki, Chem. Phys. 55 (1981) 85: a1.R. Tahcrinn, W.H. Fink and A.f!. Xlaki, J. Phys. Chem. 86 (1982) 2586. R. Arratia-Perez and D.A. Case, J. Chem. Phys. 79 (1983) 4939: D.A. Case and J. Lopez. J. Chem. Phps. 80 f 1984). to bc published. (31 [4] [5J [6] [7]
109,
CHE%llCAL
nun~ber J
RELATIVISTIC
~~OLECULAR
ORBITAL
PHYSICS
THEORY
LETTERS
OF ZERO-FIELD
3 August
SPLITTINGS
1984
IN TRIPLETS
David A. CASE L)Pparlrrrivrt
of
C~~ctrlislry.
l.hiwrSiQ*
of Caii/bnIia, DalTs. California 95616, UC?.4
:I n~cthod of valculating mnrris eien~enrs of I/r,, in B basis of IIirw scattered-wave (DSW) orbitals is outlined. In the IimiI c - cu. Ihis nwthod rcduccs to 11x11described by Cook and KarpJus for non-relativistic orbitals. For triplet states that can bc degribcd by a single confiiurntiun with two unJxdred electrons, rhe relativistic exchange intgralsgive nor only the sin_&ct --triptct splitiings (as in non-reiativistic tf~eory), but also tfre spin-orbJt contributions to rfte triplet zero-field splittings. wl’hicf~ has a wry large vahc of D fte5tIlls arc reprrrred for the 3111 * n*) excited state of ~-tliiop~~ro~c (4J~-pvr~-rl-thionc). tc;i1c. -31 WI-1, e?rp_ -23 to -28 cm-1).
1. Introduction
2. Method of calculation
Iilcctror~ic strlict ure calculations that use an effective cr;charlgc-~orrcl;tioll potcnti31 (such 3s tfje Xa or local spin density functions) can bc viewed in two difI’crc111 lights. From OIlC point of view (recc11tly revicwcd by Parr [ I]), the Xff method is an approximarion to csact density functional theory, and improvc~mnts way he possible as we gain a better understnnding of deusit y l’unct ional theory of molecular ground states. A second point of view(r~iucli closer to that hcld by Slatcr 121) views the Xa method as an zpproximate molecular orbital theory, applicable to both gr~mnd aacf excited smcs, aud capable of being itnproved by the staudard methods of quantum chemistry. e.g. through configuration interaction. In this light, Ihc Xo ii~cthod can bc vicwcd as a convcnicnt Incaus nf’gcricraliug ;Lbusis scf. for use in further calculations itI which 00 approxinmtiom are made to the f~aniiltortiatt. This was the approach adopted in early cnlculatious by Zarc [3] oi‘atomie nndtiplet splittings. and mm recently by Cook ard Karplus [4,5]. in scattcrcdwave calc~~latioris of siuglct-triplet splittings in ozone. I lerc I show how thcsc methods can be cxtcnded to the lclstivistic Dim scattcrcd-wave (DSW) urethod, and as an cXantl)Ic. report calcl)lations of the zero-field splittings in the 11-+ zr* excited state of a tltiokctonc.
The DSW method uses the smc potential approximations as non-relativistic scattered wave theory, but starts from the Dirac rather than the Schrijdinger equation. its formulation and applications have recently been reviewed [6--81. For each orbital, the wavefunction inside the atomic spheres is written as a linear combination of spinors, of the form.
60
(1) where C(&; Xktt1 = c s=+ l/2
111- s. s) Y;,,--,(0,
$?J)X(s) .
(2)
Here g(r) and fir)arc radial functions, determined by numerical integration of the radial Dirac equations 191. Y is a spherical harmonic, C is a Clebsch-Gordan coefficient. and x(s) a spii~ function, x(i) = (A), x(-i> = ((1). These spinors are cigenfunctions of J” (with cigcuvalue j(j + I)?$, where j = Ik I -+) md Jz (with cigcnvahc m?~). The orbital energies are determined by the DSW secular equations to yiefd a cont~uous wnvefunction, and the resulting orbit& transform 0 009-26 14/84/S 03 .OO 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
Volume 109, number 1
CHEhlICAL PHYSICS LETTERS
according to the “extra” irreducible representations of the molecular double point group. r As in non-relativistic theory, the wavefunction in the intersphere region is expressed 3s a multi-center combination of free-electron functions. Following earlier work [4,5,10], we approximate this intersphere part by expanding.the atomic r3dial functions beyond the sphere boundaries. This expansion accounts for all the intersphere charge, and partitions it among the atoms according to the surface charge densities in the atomic spheres. This approximation has been shown to be no worse than multiple scattering approximations themselves [ 1 I], and leads to results for molecular properties that are intermediate between minimum basis and extended basis ab initio Hartree-Fock calculations [6,1 1,121. Matrix elements of 1/rt2 can be evaluated if the operator is espanded in a convenient coordinate system. For single-center integrals, where all the wavefunctions belong to a single atom, this is the ordinary Inplace-type expansion.
The radial integrals may be performed numerically, and the angular integrals (knll YIIk’nz’) are given, to within a normalization constant, by the & coefficients of Grant [ 13,141. This procedure has been used for many years in atomic Dirac calculations, and complete details are given elsewhere [ 141. Note that the main difference from the corresponding non-relativistic result lies in the replacement of & “Gaunt” coefficients of Condon and Shortley [15] by the analogous & coefficients defined over spinors. Minor differences inc!ude replacement of Dirac for Schr6dinger radial functions, and the use of complex rather than real spherical harmonics. For two-center integrals, Cook [S] has shown how the bipolar 1/‘t2 expansions developed by Steinborn and Filter [lG] can be adapted to the multiple sc;lttering problem. Two situations arise, one where each center is an atomic sphere, and one connecting an atomic sphc~e to the outer sphere that surrounds the molecule. Use of the complex form of these expansions, along with the relativistic Gaunt coefficients, dir&It yields an algorithm for the relativistic twocenter integrals. Since the scattered-wave orbitals obey a zero differential overlap criterion, three- and
3 Augst
1984
four-center integrals do not appear. Details of the computational scheme (including the way in which various terms are stored and retrieved) are available from the author. In the limit c + 00, this scheme reduces to that described by Cook and Karplus [4] for non-relativistic calculations. 1 have reproduced their results for ozone, using a value of lOI au for c. The accuracy of these results, compared to ab initio Coulomb and eschange integrals 141, suggests that they should be useful for many purposes.
3. Zero-field
splittings
An interesting and instructive 3pplintion of relativistic configuration interaction consists of estimating the spin-orbit contribution to zero-field splittings in triplets. Consider the case in which t!ie triplet can be approximated by a (Dirac) h.10 description with two unpaired electrons. Except in very high or very low symmetry, each orbital will transform according to a 2 X 2 irreducible representation of the double molecular point group [ 171. (For the C2, case considered here, ail the Dirac orbitals belong to y5_) We cgn ad3pt a convenient non-relativistic notation to this case. using an unbarred symbol to represent an orbit31 corresponding to row 1. and a barred symbol to correspond to row 2. Hence one can construct four determinants with one electron each in orbitals a and b: Ai = labj. A, = lii61. A3 = 1361, and A3 = Ilbl. In the absence of electron repulsion, these are degenerate, but they will mix and split by the action of l/r,?. It is easy to show that A, and A2 span the B, + B, representations of Czv, whereas A3 and A3 span A, + A?. Hence the 4 X 4 CI mat*ix splits into two 2 X 2 matrices: (4) where Q = (aElb6) fl=J=g
-Jab
- (36lb5), -K,,-+k-,
.
y = (a51 6b) - (ab 1%) _
I use a standard notlltion [ 181 for the two-electron integrals. such that f.& s (aa Ibb) and K3b s (sb I ba), 67
Vllllrnlc
~llld
Illilnll~r
I 09.
;I c0111111011
cli:~gw~l
cic~llc~lts
e = +a. p 2
y
CIIINICAL I’IIYSICS LI:TTl:.IIS
I
cleIllcllt ofeq.
has
bCCl1
sublractcd
l’rolll
(4). The cigcnvnhIcs
1llC
.
-4b
(6)
(7)
lo give f) = Iyl -- (3, Ii:‘1 = ck. Ilcncc cvalualion eight rwo-clcctru~l intcgwls of cq. (5) directly
of tlic
to a thioketone the /.cro-field splittings
not by spirl-orbit
coupling,
but rather
by spin-spin dipolar in1cracrions. which are no1 includcd in the theory prrscutcd above. RccerItly it has been found that ccrrain II - 7r* triplet sIaIcs, cspecially those conIairiiIig a sccorid row atoni. can have valucs of I> greater tluui 10 cIIi~- I. illi order of magnitude larger than typical spin-spin interactions. One ofthe simplesr of Ihcse is 41 I-pyran-4-thione (PT. fig. I ), for \vliicll II vulues of -24 and -28 cu- I have been obw vcd for two distiiw trap sides [20]. X conventional non-relativistic Xck multiple scatIcriug s:~lcula~ion ]2 I ] on I’T gives t hc rcsults shown iu fig. I. TIIc geonicIry was that found in a recent niicI-owave study ]22]. and sphere radii were chosen by Illc Ko1ni311 procedure 123 J with a scale factor of 02%. Partial waves through I = 2 were included for the ouler sphere and S, Ihrough I = I for C and 0. and I = 0 for I I. The two highest occupied orbitals are Gb, (priniarily loue pair, with 33% of the charge associated with the p_,.orbital on sulfur) and 3bI (a ITorbital with about 702 occupation on S). The lowest unoccupied orbital
is 3b t (E* with about
Upon going to a Dirac calculation, 68
PT
2 z -6E El
_6b2 3”, -lo
2
yields
Ilic /em-field spli11iIigs. II is worIlIwhilc to cwnparc this to the corrcspondiug mm-relativistic calculation. There a and ii have Ihc sa111cOIbiral part aud differ only iu their spin coulpo~wuts. In tlIis case it is easy to set that o = 0 and 0 = I-yl. so tht ccl. (6) yields three degcncrate triplet suhlcvcls and a siuglct separated from thcni by ZK,,,. Tlw ~ffcc~ of I IIC spin-orbit tern1 implici1 in the Dirac I IauIiltoui:III is 10 rcuiove this triplet state dcgcncracy.
For many organic triplets,
-4-
l
WHY,&= f>(SJ -- $.s’, + /fc.s~; -- s>:,
31-c don~htc’d
b
arc
III Ihe prcsc111 case (see below), lol Q /3 = lrl, and the three lowest eigcnvalucs cau bc fit to a conventional spin I13111i11011i311 [i’,j
4. Application
3 August 1984
10% S character).
there is significant
-8 -2b
t
spin-orbit mixing betwccu 6b2 and 3b,, yielding a top occupied orbital which is primarily sulfur lone pair, with sonic n cltaractcr ntixcd in. Shnilar behavior is seen if the non-relativistic calculation is converged with a single electron in bb, and 4bI (corrrcsponding 10 a misturc of I(n -+ n*) and 3(n + n*) sIaIcs). It is the potential from this latter calcuhuion that was used to obtain the results reported below. Alrl~ougl~ Dirnc spinors have four components, for the purpose of qualitative analysis we can ignore the bottom two. “snIall” conlponeuts, and identify the top two components with spin Q and /3. If WC further assun~ that the radial functions of thcj = I + i and j = I - i spinors arc the same (an excellent approxiniaIion for light atoms) we can decompose the Dirac orbitals into a I’auli form consistiug of (real) spherical IIarnIonics Inultiplied by spin functions. The results of this deconiposition are shown in table 1 for “row 1” of the irreducible representation ys. (Row 1 is defined to transforIii in the SdlllC way as a spin function with j =- i, m = -4 [ l7]-) Since both the direct product (b 1 X CC)and (b, X a) belong to row I of rs, the Dirac orbitals are nearly pure Q’spin states; small admixtures of a1 and a2 orbitals provide some 0 occupation_ The real part of the a function (component 1) corresponds to the lone pair (b?) orbital, and the imaginary part belongs to the b, (nj orbital. As table 1 indicates, the lower partially occupied orbital (which 1 call “a”) is mostly (91%) sulfur lone pair with some a character. (The ratio of lone pair to a coefficients on sulfur is
Volunic 109. number 1
3
CIiEhlICAL l’I1YSICS LETTERS
Aug11sr
1984
Table 1 Orbital characters Pauli decomposition -
Orbital a(“6b,
“) =
b(“4b;‘)
=
0.9518 -0.0595 +0.1363 -0.0027i + SIWlhx
s s Cy S
+ + -
0.0173i 0.05683 0.0165i 0.003Si
S S Cy S
I’.\ d,=
Q Q
1,:;.11~
a Q n D
!‘S d,2
;
I’s d AZ I’s I’.\I’s
a 0 Q * (r
+
O.OOlSi O.OOOli 0.0001 i 0.0003i 0.0002i
S S Cy Ca 0
i’_J $*z I’?’ I’_V P.1’
Q Q Q Q 0
p,. d.-
terms
-0.2090 s +0.3012 s -0.6282 Cy +0.4100 Ccu -0.3339 0 + s111a1kr tcnns
_-
55 : 1). The higher partially occupied orbital (“b”) is of 77character with a very small amount of in-plane character (a coefficient ratio of 150 : 1 for the sulfur p-orbitals). (Interestingly, the dr occupation on sulfur in orbital b is even larger than the prr occupation; I plan a future study on the effects of sulfur d-orbit& on thioketone zero-field splittings.) The relevant two-electron integrals arc given in table 2. The value of D is determined primarily by the difference between Kab and (ab Ii%), whereas E is detcrmincd niainly by (a6lbT). An esmination of
Table 2 Two-electron
integrals a)
( 1) non-relativistic b) J,,*=51194 J nn* = 45731
%rrl * = 9544 Knrr* = 631
(2) relativistic Jab = J& = 44273 c) (a:lbG)= -2.5 X 1O-7 (alllgb) = 2.3 X 10m5 (3) zero-field splittings d) D talc. = -31.5 I./I?!talc. = 0.26
&,t, = 428.3 (abli%)= -396.8 (a61 b?i) = 0.26
tsp. = -28. -14 esp. = 0.060. 0.063
a) All values in cm-‘. b) Dcsigations x, IT* and n rever to 3bt. 4bt, and 6b2, respectively. c) The equality of Jab and Jag is a consequence of symmetry in Czv; these integrals may not be equal in other point groups, such as C3v or Dgh. d) Esperimental values from ref. [ 101.
+ -
t lie C,,
COUpliilg coefficients [ 171 shows that, for the non-relativistic orbitals, K,,* = (mr* Iii*%). whereas A’,,,,+ and (nn*lii*ri) are equal inmagnitude. but have opposite signs. Hence a mixture of II and in clraracter in orbital “a” gives a &, not equal to (ab It%). Since the mr* exchange integral is an order of magnitude larger than that for nn* (see table 2), even a small admhrure of 71character into a lone pair orbital can yield significant zero-field splittings. It is not unreasonable to expect that other thioketones may exisr with even hrger values of D. and the observed sensitivity of D to the trap site is easy to understand in terms of very small changes in the amount of n-r mixing. Our value of E is larger than the experimental one by about a factor of four. This apparently arises from an overestimate of mixing to the Ia? (sr) orbital, which destroys the spin symmetry that otllerwise would make (a6lbZ) zero. In spite of this error, the theory does predict a value of E two orders of magnitude smaller than D, in accord with experiment. A similar calculation on the 3(5r + x*) state (with configuration 3b 1 + 4b 1) gives D = -0.85 cm- I. Although this value is not known experimentally for PT, it is in rough accord with observations on other thioketone n--71* states, such a thiouracil [24]. Thus the simple single configuration theory presented here appears capable of reproducing the large difference seen between 3(x +- .rr*) and 3(n + 7r*) states.
5. Discussion One principal
advantage
of the Dirac Hamiltonian 69
CHEMICAL PHYSICS LETTERS
Volume 109, num bcr 1
is that the spin-orbit interaction is contained in the zero-order states, and does not need to be added as a perturbation. This reduces the order of perturbation theory required to calculate properties that involve ;~I~quer~cl~ed orbital anguIar ~~~o~~~entu~~~. The usual theory [ 191 of the spin-orbit contribution to zerofield splittings involves a second-order perturbation stun over all cscited states. This can be very difficult to carry out since reliable infomlation about the nat urc and locarion of molecular excited states is rare& available. Such sun~s are not required in the theory presented here. Earlier [ZS] WC have used a similar approach t0 develop first (tather than second) order perturbation theories of orbital contributions to IIIOlecular hyperfine interactions. The calculations presented here can be extended in 1tv0 ways. First, spin-spin interactions can be included vh the Brcit operator. Expressions for the required one-center matris elements (which should be the nlost intlx)rtanl ones) have been dcvclopcd for use in atoniic calculations [ 141, and could easily be incorporated here. Second, the triplet state wsvefunction could be improved through the addition of nlore configurations. This would require the calculation of niatrix elements ot’one-particle 3s well as two-particle interactions. As with the 1/rll operator, this c;fn be accon~plished by a rclativcly straightforward nwdificstion of cxisring non-relativistic scattsred wave codes. This opens the
w:*y to 3 gencr31 Ci code for DSW wavefunctions: p10gran1s to carry this out ltre currently
under
de-
Vclopnlent. I:vcn wit/lout thcscextcnsions. however. the present results suscst that the sin~p!c theory here should bc uscii11 as a qualitative guide to zero-field splittings in
mhxules
with large spin-orbit
internetions.
Since
inan?; ursanic
arid transition mct;tl complexes I>11 in%0 llG.5 category. a large number of interesting spectroscopic q~plications can be cnvisioncd.
Acknowledgement
I Ihmk A.1 1. hlaki for pointing 0uL tfic interesting bcltsvior of PT. ;tnd ht. Cook for providing n copy of his non-relativistic codes. which were rhc basis for those described here. This work was supported by the National 3n Alfred
70
Science Foundation, I’. Slosn
Foundalion
CHE-!31-17101. Felbw.
DAC is
3 August 1984
References [l] R.C. Parr, Ann. Rev. Phys. Chem. 34 (1983) 631. [ 21 J.C. Slater. The self-consistent field for molecules and solids (hicCrawHil1, New York, 1974). R.N. Zarc, J. Chem. Phys. 45 (1966) 1966. M. Cook and M. liarplus, Ctlem. Letters 84 (1981) 565. ill. Cook, Ph.D. thesis, Harvard University (1981). D.A. Case, Ann. Rev. Phys. Chem. 33 (1982) 151. C.Y. Yang, in: Relativistic effects in atoms, molecules and solids, cd, G. hlalli (Plenum Press, New York, 1983) p. 335. (S] C.Y. Yang and D.A. Case, in: Local density approsimation: in quantum chemistry and solid-state physics, eds. J, Dahl and J.P. Avery (Plenum Press. New York, 1984) p. 643. [ 91 J.E. Ilarriman. Theoretical foundations of electron spin resonance (Academic Press. New York. 1978). [ 10 ] D.A. Case and 51. Karplus, Cbem. Phys. Letters 39 (1976) 33. [ 11 f $1. Cools and Xi. Karptus, J. Chem. Phps. 72 (1980) 7. [ ‘It ] D.A. C;tse, hf. Cook and M. Karplus, J. Chcm. fhys. 73 (1980) 3294. [ 13 1 I.P. Grant, Proc. Roy. Sot. A262 (196 1) 555. [ 141 I.P. Grant, Advan Phys. 19 (1970) 747. 1IS] E.U. Condon and G.11. Shortley, The theory of atomic spectra (Cambridge Univ. Press. London, 1935) ch. 6. [ 16 J E.O. Steinborn and E. Filter. intern. J. Quantum Chem. s9 (197.5) 135. 1171 G.F. Kostcr, J.O. Dimmock, R.G. \\‘bceler and 11. Statz, Properties of the 32 point groups (MIT Press, Cambridge. 1963). R.G. Parr, Quantum theory of molecular electronic structure (Benjamin, New York, 1964) section 5. N.%f_Atherton, Electron spin resonance. theory and applications (Wiley, New York, 1973) ch. 5. h1.R. Taherian and All. Maki. Chcm. Phys. Letters 96 (1983) 541:Chcnt. Phys. 68 (1982) 179: D.11. Uurlnnd. J. Chem. 1’11~s.75 (1981) 2635. Program No. 465 from Quantum Chemistry Program Exchange, Bloomington, Indiana. J.N. MacDonald, %A. %facKay, J.K. Tyicr, A.?. Cos and 1.C. ilwart. J. Chem. Sot. Faraday Ii 77 f 1981) 79. J.C. Norman Jr., Mol. Phys. 31 (1976) 1191. M.R. Taherian arId AM. Maki, Chem. Phys. 55 (1981) 85: a1.R. Tahcrinn, W.H. Fink and A.f!. Xlaki, J. Phys. Chem. 86 (1982) 2586. R. Arratia-Perez and D.A. Case, J. Chem. Phys. 79 (1983) 4939: D.A. Case and J. Lopez. J. Chem. Phps. 80 f 1984). to bc published. (31 [4] [5J [6] [7]