SPIN-FREE QUANTUMCHEMISTRY
2477
be offered for this sort of relationship. Further work on similar isotope exchange systems involving different transition metal complexes is in progress.
Spin-Free Quantum Chemistry. VI.
Acknowledgment. The authors wish to express their sincere thanks to Dr. J. Shankar for his continued interest and encouragement during the course of this work,
Spin Conservation1
by F. A. Matsen and D. J. Klein Molecular Physics Group, Department of Chmistry, The Uniuersity of Texas, Austin, Texas
(Received June 11, 1968)
Many chemical systems are well described by the Breit-Pauli Hamiltonian with the spin-free term and spin terms treated as zero-order and perturbation Hamiltonians, respectively. If the zero-order levels are widely separated and the spin effects are small, a system admits, to a good approximation, a spin-free formulation. Since the spin-free Hamiltonian commutes with the group of permutations on spatial electron coordinates, the partitions [A] which label the irreducible representations of this group are good quantum numbers and label the spin-free states. In this regime [A] is conserved in collisions, chemical reactions, and electric dipole radiative processes. The Pauli principle restricts the physically significant permutation states and establishes a one-toone correspondence with spin states, It follows that in the spin-free regime spin is a good quantum number and is conserved. Rules are derived for the conservation of permutational quantum numbers between separated and composite systems. The conventional spin analogs of these rules are the Wigner spin conservation rules. We discuss two types of breakdown of the spin-free permutational symmetry: (i) the breakdown of local permutational symmetry while preserving total spin-free permutational symmetry (an example is the enhancement of singlet-triplet transitions by collisionwith paramagnetic molecules); (ii) the breakdown of total spin-free permutational symmetry. The breakdown is practically complete in those cases for which the zero-order states are degenerate or near degenerate. A number of photochemical processes are discussed with particular reference to methylene and benzene.
I. Introduction Many atomic and molecular systems are well described by the Breit-Pauli Hamiltonian
+D
H = HSF
(1.1)
where contains all spin terms and HSFis spin-free. The states of the Hamiltonian H are characterized by a rigorously exact total quantum number which we designate by K . (Such exact quantum numbers may &, or F ( F = or anirreducible be J ( 7 = representation of a double point group, or just the energy sequence of the states.) If the matrix elements over D are small ( e . g . , Russell-Saunders coupling) one can apply perturbation theory taking HsF as the zeroorder Hamiltonian and $2 as the perturbation. The Pauli-allowed portion of the total Hilbert space of the Hamiltonian H of (1.1) is
z+
+ s),
w = a(VSF63 VU)
(1.2)
where V u is the fermion spin space, VsF is entirely spin-free, and a is the antisymmetrizer. The spin space V u may be decomposed into invariant subspaces with respect to the symmetric groups SNuof permutations on spin coordinates
where the partitions [A"] identify irreducible representations of SNu. Only partitions of the form
[XU] = [N
- P , PI
(1.4) may appear in (1.3), since there are only /a)and I@) spin orbitals. The partition [N - p , p ] determines the spin quantum number 23 associated with the space V ([A"]) through
N
s = -2 -
P
(1.5)
The spin-free space TisF may be similarly decomposed
VSF =
c VS"([P])
[W
(1 6) Q
The subspaces VsF( [ASF]) are invariant with respect to the permutation group SNsFon spatial coordinates. Substituting the decompositions (1.3) and (1.6) into eq 1.2 for W and using the antisymmetrizer (as given in eq A.8) we see that (1) This research is supported by the Robert A. Welch Foundation of Houston, Texas.
Volume 78, Number 8 August 1969
2478
F. A. MATSENAND
w = a xI V S F ( K ) 63 V'([X"])
(1.7)
[AU
3
where is the conjugate partition of [A" 1. only [AsF J which survive are of the form
[xSF] =
/sJ
[A"]
1-
Thus the
= [2p,lN-2p](1.8)
=
These partitions in (1.8) are called Pauli-allowed partitions. From (1.7) it is apparent that confining a spin ket of V u to the permutational symmetry [A"] = [N - p , p ] implies that the corresponding spatial ket in an antisymmetrized spatial-spin lret is of spin-free permutational d symmetry [ASF] = [Xu]. A ket of pure permutational symmetry [ 2 p , l N - 2 p ] will be contained entirely within /v
F([AS")
63 VU([ASF]>. The zero-order Hamiltonian HSF operates only on VSF. Since
[HS",P] = 0,P E S,SF
(1.9)
the partitions [XsF] become the zero-order quantum numbers which label the zero-order states. We call these zero-order states permutation states and the [hSF] spin-free permutation q u a n t u m numbers. Those systems to which the zero-order theory applies are described in a spin-free f~rmulation.~-?Since in a spin-free formulation only SNSF and [ASF] are used, we will suppress the superscript SF. I n the spin-free formulation2-' one p a y begin with a basis
B(u) =
{I"$);
i
=
1 to
rf
(1.10)
To obtain an a priori labeling of the permutation states one forms a symmetry adapted basis B(uS) ZEZ
c [A]
jPl
~lv.;[X]r); r=1
7
= 1 to.PAlj
[X,P] = 0, P E 6°F
For any
=
N E r$,the dipole moment operator, one obtains i-1
the intercombination rule. Thus [AsF]is a good quantum number and is conserved in radiative electric dipole transitions between spin-free states. Similarly, when other chemical processes, such as internal conversion, collision, dissociation, or isomerization are well described by spin-free transition operators, [XsF] is again conserved. By (1.5) we see that spin S is a good quantum number and is conserved in these same processes. Further, as a consequence of eq 1.7 even for a system with a Hamiltonian possibly containing spin interactions: s p i n i s conserved i f and only if spin-free permutational symmetry i s conserved.
TI. Breakdown of Local Permutational Symmetry I n this section we discuss systems in which permutation symmetry appears to be broken but is in fact conserved. Let A and B denote two molecules in a two molecule system which admits a spin-free description. We write the spin-free Hamiltonian as
HSF = H A
+ H B + HAB
(2.1)
where HAand HBare the Hamiltonians for the isolated molecules A and B, and H A B contains all remaining intermolecular interactions. The molecule AB may be treated in the vector space
VAB=
P ( V A €3 V")
generated from product kets
IAB;W)
E
IA;KA[XA]rA)
€3
/B;KB[XB]TB)
(2.3)
Here the kets in VA are associated with electrons 1, 2 , . . . , N A , and those in V" are associated with electrons NA 1, NA 2 , . . . , N A NB. We choose IA;KA[kA]rA) and (B;KB[xB]TB) to be eigenkets to H A and H B in vector spaces VA and V", respectively, and to have eigenvalues Ea(A;KA [AA]) and E B ( B ; K B [XBI). There are bases for TiAD which are symmetry adapted
+
+
+
(2) (a) F. A. Matsen, "Advances in Quantum Chemistry," Vol. I, P. 0. Lowdin, Ed., Academic Press, New York, N. Y., 1964; (b)
F.A. Matsen, J. Phys. Chem., 60,3282(1965).
(1.13)
[ ~ l r j"7'; ~ l[ ~ ' l r ' )=
6([AI, [~'1)6(~,r')(~~;[~l(~X~~~~';[Xl) (1.14) The Journal of Physical Chemistry
X =
P@N
The Special Wigner-Eckart theorem holds (UT;
where (uT;[A]//XI/VT';[A]) is a reduced matrix element independent of r and r'. As a consequence ofQ.14) with X = HsF, the secular equation is broken into' block diagonal form, each block uniquely characterized by particular quantum numbers [XI and r. We denote the eigenvalues of this secular equation by E(u;K[XI) and its eigenkets by lu;K[A]r). If in (1.14) we choose
(1.11)
related to B(v) by a basis transformation. Here the symmetry adapted basis ket I u 7 ; [hlr) transforms as the rth row of the [Xlth irreducible representation of SNsF, T distinguishes between different basis kets with the same [ X I and T quantum numbers, and fUlx1 is the number of times the fIX1-dimensional[hlth irreducible representation of SNsFoccurs in the (reducible) representation r"generated from the basis Buin (1.10)
where x['](P) and x"(P)are characters. operator X such that
D.J. KLEIN
(3) F. A. Matsen, A. A. Cantu, and R. D. Poshusta, ibid., 70, 1558
(1966). (4) F.A. Matsen, ibid., 70, 1568 (1966). (5) F.A. Matsen and A. A. Cantu, ibid., 72,21(1968). (6) F.A. Matsen and A. A. Cantu, ibid., 73,2488 (1969). (7)F.A. Matsen and M. L. Ellaey, ibid., 73,2495 (1969).
SPIN-FREE QUANTUM
2479
CHEMISTRY
to 8,; m h symmetry adapted basis kets may be projected from the product kets of (2.3).
JABos;[X]r}= (Normalieation)e,s[x~~AB;o}(2.4) Here erS[’]is a group theoretical operator called a matric basis element (see Appendix) and is such that the resulting ket (2.4) transforms as the rth column of the crth irreducible representation of SN. We note that some of the terms (2.4) are identically zero for group theoretical reasons. The product kets !AB;”) transform as 8 r[XB1 an irreducible €3 S N B . This irreducrepresentation of the group ShrA ible representation of S N A @ SUBinduces* an outer direct product representation I’[xA1@[[XB1 in the group SN. The representation r [’A1c3[AB1 is reducible
translated into the conventional spin formulation through the equivalence (1.5)- A number of examples of the decomposition (2.5) or (2.10) of outer direct product representations are given in Table I. Table I : Spin-Free Wigner Spin Conservat’ion Rulesa Molecule A
Molecule B
Composite AB molecule
[SA] [SA] [SA] [SA] [DAI [DA] [DA]
[SBI
[SI [Dl
[TA]
[TB]
[TA] [&A]
[DBI [TB]
[TI
[Qd
IQ1
[DBI
[SI [Dl
[TB]
0 [TI @
[&I
0 [&’I [SI 0 IT1 0 [&’I [Dl 0 [&I 0 [S’l 0 IT1 0 IQ’I @ 1S”l [TI
[&B]
[&I
[&B]
[SI
Note: the partitions [SI [Z’], [D] E [2’,1], [TI 3 [2”,1’], [Q] E [2”,1*], [Q’] = [2*,14], [S‘] [2*,16],and IS”] E [2”,le] represent singlet, doublet, triplet, quartet, quintet, sextet, and septet states; similar notation is used for the [XA] and [AB] partititions. a
The corresponding dimension statement is
=
(2.6) For [A,] (2.5)
=
[AB]
= [2, 11 we have a specific example of
The eigenkets of the total Hamiltonian (2.1) in VAB may be expressed in terms of the kets (2.4) which are symmetry adapted to S N .
C IAB;ws[AIr}(wslK[X]r}(2.11)
IAB;K[X]r) = w
s
In the limit of large intermolecular separation there are particular local quantum numbers such that for many operators X lim (AB;K [A]rlXIAB;K’ [h’lr’) = m
(A;KA”[XA” ITA~IXAIA;XA~’ [XA~’]TA*‘)
(2.7)
Thus a product ket symmetry only if f[AA1’[’B1 f[xAl@[’B];[’]
1AB;o) has a component of ;[’I # 0 , that is
= Q =
e,,[’]IAB;~)=
Q
(2.8) Given [A,] and [AB], this requirement (2.8) restricts the partitions [ A ] which may arise. For the Pauli-allowed exactly one each arise for partitions [ A ] = [ZP, 1N-2P] which N - 2p is such thatg
(NA
- 2pA) + (NB - 2pB) 2
1(NA
- 2p.4)
(N - 2 p ) 2
- (NB - 2pB)/
(2.9)
We write this result as r[’Al@[’’]
Pmax I
+
r[29,1v-2~1
P = Pmin
Pauli-excluded partitions
(B;KB” [hBm]l”BmlXBIB;KBm’ [hBm’]rgm’)
+ (2.12)
Since WsF is one such operator, (2.11) becomes lini 1AB;K [X]r) R-tm
(ABwms;[XIr}(w”s~K [X]r) (2.13)
= 9
Another operator X which satisfies eq 2.12 is the -+ dipole moment operator r. Thus we see in the limit as R -+ m , the local quantum numbers w m = (Ka“ [XA“]~ A ~ , K B ~ [ X} are B ~exact, ] ~ B ~while for any finite R they are not exact. For finite but large R or for weakly interacting molecules w m may be expected to be a good approximate quantum number. It follows from the above discussion that in a nonadiabatic spin-free collision (Le., [ X I fixed) between two molecules A and B a change in local permutational symmetry may take place
(2.10)
where p m i n and p m a x are defined by the limits of eq 2.9. This is a spin-free derivation for the Wigner spin conservation rule. The results obtained here may be
(8) G. de B. Robinson, “The Symmetric Group,” University of Toronto Press, 1961. (9) The relations (2.8)and (2.9) follow from the graphical technique described by M. Hammermesh, “Group Theory,” Addison-Wesley Publishing Co. Inc., Reading, Mass., 1962, pp 250-252. Volume 73, Number 8
August 1969
2480 A([AAI)
f B([hBI)
-
I?. A. MATSENAND D. J. KLEIN f B([hB'I) (2,141
A([AA'])
Before collision the bimolecular system AB has approximate local permutational symmetry [XA] and [AB]. After the collision the sysbem AB can have different local permutational symmetry [XA'] and [AB'], which are restricted by (2.9). Additional restrictions on [XA'] and [AB' ] are oftpen obtained on considering the energetics and dynamics of the system. Many examples of collisions with changes in local permutational symmetry are known. The Cundall technique'' for measuring triplet yields of organic molecules involves the transfer of triplet state excitation energy
C,IH@BI~) 4- 2-butene([S]) -3 C~H,I(~AI,) f 2-butene( [TI) (2.15)
This behavior has been explained16- 18 without recourse15 to the influence of the magnetic field of the paramagnetic species on the electron spins of the organic molecule. In paper I1 of this series,2ba spin-free formulation of the paramagnetic effect was presented. We now repeat the spin-free formulation utilizing the notation of the present paper and the concept of local permutational symmetries. We consider the system to consist of one organic molecule A with low-lying states [SAo], SA^], and [TAO]and one paramagnetic molecule with ground state [TBO]. Typical energy levels of such a system are shown in Figure 1. Within the approximation of considering configuration interaction only between the [TA'] C3) [TB'] and [SA'] 8 [TB']separated molecule configurtions, the triplet eigenkets of the AB system are 1
The triplet-triplet annihilation mechanism" for intersystem crossover explains the quenching of phosphorescence of organic mo1ecules12J9by oxygen
1 [T21)= 1/1+,.2(-al
C~HO(~B 4-I 0~2 ()3 & - )
1 [TI]) = ~
4CeHs('Aig)
4- Oz('Ag) (2,16)
LTAo1
[TBol];[Tl)
1 [SA'] [TB']; [TI))
(3 * 1)
1 ~ ( ( T A[TJ3' ' 1] a([SA']
[TBO]
(3 2,
In each of the collision processes indicated in (2.16) and (2.17) we note that other products do occur, and that in certain cases adducts or compounds may arise. I n both these reactions, (2.15) and (2,1G), the total [ X I is conserved. Equivalently, using (1,5), the total spin S is conserved.
(3 3) where we suppress K and Y. The mixing coefficient a is then
111. Effect of Paramagnetic Molecules on Spectra It has been ~ b s e r v e d ~ ~that J * J paramagnetic ~ species, as 02,NO, or certain metal ions, often enhance permutationally forbidden transitions in organic molecules.
A E E[SA']- E[TA']
1 [To])=
@AB)
5
[SAoIITBol;[Tl)
(3 5) '
([SA'][TB'];[T]/HABI [TA~I[TB~I;[TI) (3.6)
Within the above approximations the transition dipole moment between [TO]and [T'] is ([T'l/i.l [T'I)
=
I
Separated Molecule Limit
Bimolecular Composite System
Figure 1. Unperturbed energy levels of the organic molecule A and B in the separated molecule limit and the corresponding perturbed energy levels of the bimolecular collision complex. The Journal of Physical Chemistru
(10) (a) R. €3. Cundall and D. G. Milne, J. Amer. Chem. Soc., 63, 3902 (1961); (b) R. B. Dundall, F. J. Fletcher, and D. G. Milne, J. Chem. Phys., 39,3536 (1963). (11) H. Sternlioht, G. C. Nieman, and G.'W. Robinson, ibid., 38, 1326 (1963). (12) G . Porter and M. W. Windsor, Proe. Roy. Soc., A245, 238 (1958). (13) K. Kawaoka, A. U. Khan, and D. R. Kearns, J . Chem. Phye., 46,1842 (1967). (14) (a) D. F. Evans, J. Chem. Soc., 1987 (1967); (b) G. Porter and M. R. Wright, Discuseion Faraday Soc., 27, 18 (1959). (15) (a) D. F. Evans, J. Chem. Soc., 1351, 3885 (1957); (b) D. F. Evans, Nature, 178,534 (1957). (16) G. J. Hoijtink, Mol. Phys., 3,319 (1960). (17) H. Tsubomura and R. S. Mulliken, J. Amer. Chem. Soc., 82, 5966 (1960). (18) J. N. Murrell, Mol. Phys., 3,319 (1960).
SPIN-FREE QUANTUMCHEMISTRY
248 1
S is not a good quantum number either, Thus the eigenkets of (4.1) are of mixed permutational symmetry. As an example, we take a system composed of two spin-free states, [SI and [TI, which are mixed2"by the spin interactions 0. It is assumed that the spin-free energies E o [ S ]and Ea[T] are functions of a system parameter Q, e.q., an internal nuclear coordinate, such that the spin-free levels cross as shown in Figure 2. The eigenkets and eigenvalues for the Breit-Pauli Hamiltonian are
Assuming that near the limit of large R the matrix element (HA=) in the coefficient a approaches zero more slowly than ( [SAo][TBo];[T]/;( [TAoI[TBoI;[T]), then for large R ([Toll;l [T'I)
-
Using (2.12) and assuming that the dipole moment of B is zero, as for 02,we obtain for the transition from [TO] [TI1
+
E(I1) = '/2(E0[T] I n this formulation the enhancement of what appears to be a singlet-triplet transition borrows intensity from the singlet-singlet [SA"]--t [SA'] transition. Thus we have a spin-free mechanism for apparent singlettriplet transitions. An example is provided when the organic molecule A is benzene and the paramagnetic molecule B is oxygen Cr&('Aig)
I
5
+ 02('Zg-)
+ Eo[S]+ Y)
and where
EC~H~.~Z('AI,~~,-;[T])
J +hu
+
Y =d A 2
C7H6.Od3Blu3Zg-;[TI) (3 lo)
CeH6(3B~u) Od32,-)
+ 4(Q)2
(4.11) (4 * 12)
I
A process of similar nature may account1s for the quenching of triplet He in electrical discharges
+
H e ( l ~ 2 s , ~ S ) He+(ls,%)
kJ.
He2+(%+)
J -hu
He(ls2,%)
+ He+(ls,*S)
He2+(?Zg+) (3.11)
1V. Breakdown of Spin-Free Permutation Symmetry The eigenkets to the Breit-Pauli Hamiltonian are of the form
jx)=
(4.6)
KSF [W]
IKSFIXSF])(KSFIXSF]IX) (4.1)
where K is a total quantum number and where /KsF[ASF]) is an antisymmetric space-spin ket (see Appendix).
The matrix element (G) which mixes permutational symmetries is in general nonzeroz1unless required to be zero by reason of the double point group22symmetry. With (a) # 0, the energy levels and extent of mixing are shown in Figures 2 and 3. It is assumed that A is strongly dependent on the system parameter Q and that ( Q ) is relatively constant under changes in Q. (Q = QO at A = 0) For Q < Qo, A >> 0, a En[T],and 0, E(1) [T][S];I) % [TI). For Q > Qo,A &a
Such a process is called an intersystem crossover, since the molecule changes from triplet to singlet. We may also J K ~ ~ [ [ X ~ " ] ~ ) represent such a process in terms of electron pairs d
IKSFIXSF])= (Normalization) a@") @I /ill[hsFJr}) =:
(il'ormalization)
C r
w
/Ill[ASF]l.) (4.2)
Since
[H,SNSF€3 B ] # 0
(4.3) [ASF] is not a good quantum number. Analogously, since [H,S €3
321 #
0
(4.4)
(19) J. C.Browne, private communication. (20) Although there are three degenerate spin-free [TI states M = -1, 0, +1, we need consider only one ket from this [TI space. All three [TI states are considered in the discussion attending Figure 5. We also neglect the diagonal part of the spin interaction. (21) J. yon Neumann and E. P.Wigner, Phys. Z., 30,467 (1929). (22) (a) D. €3. McClure, J. Chem. Phys., 17, 665 (1949); (b) S. I. Weissman, ibid,,18, 232 (1950); (0) F. A. Matsen and 0. R. Plummer, "Group Lattices and Homomorphisms" in "Group Theory and Its Applications," E. M. Loebl, Ed., Academic Press, New York, N. Y., 1968. Volume 73, Number 8 August 1969
-
2482
MZ, -+MI
M: M:-w+ or M 3
Q C Qo
Q
F. A. MATSENAND D. J. KLEIN
(4.14)
Q > Qo
Qo
Here the twci equivalent symbols M:+-+Mz, and M 2 represent resonance hybrids between the singlet M 3 and triplet M : forms. Another notation for this intersystem crossover process is obtained if one denotes the singlet [SI by TJ and the triplet [TIstate by 77
-
Q
It SC:
-M-+
Qo
77
-
Q i~
M --*M
(4,15)
Q > Qo
Qo
Here tt 11 represents a resonance hybrid. Representation of the intersystem crossover as in (4.15)gives rise to the deiicription of this process as a spin-& process. We note that the term “spin-flip” may be misleading. 2 3 We see that for this intersystem crossing process permutation quantum number [ASF] is not a good
quantum number and is not conserved. Correspondingly, S is not a good quantum number and is not conserved. The rate at, which an intersystem crossover occurs is a function of the shape of the potential surface, the matrix element (a), and the characteristics of the initial wave fiinctions of electronic and nuclear motion. Often one expects the rate of crossover to be a maximum when the kinc4iic energy of nuclear motion in the region of the zero-order crossing is near zero. Such nonradiative intersystem crossover is often referred to as the Landau-Zener effect.24 We note too that even though there is no crossing in the perturbed potential curves, certain prwesses, as high velocity collisions, may in effect act as though there were no perturbation and thus follow tlw zero-order potential curve and conserve [XI. Thus an alternate type of intersystem crossing process may occur
I[Tl)---+ j[‘~I[Sl;I)-+
Q
Q
&a
l[Sl[Tl;IJ)----t Qo
Q
as a function of A or &. The Journal of Physical Chemistry
(23) I n elementwy discussion, a magnet with two and only two orientations is ts.ken as a classical analog of the spin of an electron. I n this model the orientation of one of the spins must flip in the transition I [TI)-w+ 1 [SI). Since the elementary magnet can take one of only two orientations, the model suggests that the spin flip occurs at a precit,e value of Q or for slow reactions at a precise time. In general classical models of spin and spin conservation may break down. The so-called spin operators which occur in the BreitPauli equation result from the Dirac formulation which does not make an analogy to classically spinning particles. (24) (a) L. D. I,rtndau, Phys. Z h . Sowjet., 2, 46 (1932) ; (b) C . Zener, Proc. Roy. SOC.,A137, 696 (1932); (0) E. G. C. Stueckelberg, Helv. Phys. Acta, 5, 369 (1932) ; (d) C. A. Coulson and K. Zalewski, Proc. Roy. Soc., A268,437 (1962).
2483
SPIN-FREE QUANTUM CHEMISTRY example, there is evidence26v28that the lowest lA1 and 3 B curves ~ have a zero-order crossing at an HCH angle of about 125’. In the presence of spin interactions the noncrossing rule applies so that the perturbed potential curves appear as in Figure 5. I n Figure 5 the potential curves are labeled by their symmetry in the appropriate double point group.22 Thus we may expect intersystem crossing of ‘A1 methylene to %BImethylene, expecially for a vibrational level of ‘A1 near in energy to the zeroorder crossing. In the flash photolysis of diazomethane2s high pressures of K2 give high yields of triplet [TI methylene, and low pressures of K2give low yields of [TI methylene. Since CH2 is expected to be initially formed in singlet [SI state, the presence of Nz appears to enhance intersystem crossing. There are a t least two mechanisms by which collision with NZ might induce intersystem crossing. First, in a plot of potential energy VS. CHz-Nz intermolecular separation, there may be a zero-order crossing of [SI and [TI potential curves. Then following (4.13)
Energy
I
L HCH Figure 5 , Behavior of the perturbed energy levels near levels in methylene. the zero-order crossing of LA1 and 8B1
by some other molecule which scavenges vibrational energy. Intersystem crossover has been found2’ to relate the
~
where Q is a CHz-S2 intermolecular separation. Alternately the Nz molecule may serve only to remove excess vibrational energy from an excited singlet electronic state of CH2. Such rapid vibrational relaxation could serve to enhance intersystem crossing in a situation as depicted in Figure 6. In this second mechanism we imagine process (iv) of Figure 6 to dominate in the absence of vibrational relaxation, and the processes (i), (ii), (iii) to dominate in the presence of rapid vibrational relaxation. The first mechanism will not necessarily occur if Ns is replaced by some other molecule, since the intermolecular potentials may be different. The second mechanism however, is expected to occur whenever Nz is replaced
QcQo
Q i Q o
Figure 4. Transitions from a lower lying singlet state \[So]>to the perturbed stat,e /IT] [SI ;I>.
Q’Qo
fluorescent and phosphorescent yields in Cr3+ octahedral complexes to the location of the crossing of zeroorder 2E and 4T2curves. This adiabatic intersystem crossover also explains certain predissociation phenomena. We also note that the crossing of zero-order aTz and lE curves in Ni2+ complexes accounts for** an enhanced spin-forbidden transition. The effect of spin-orbit splitting on spin-free ligand field curves has been discussed and calculated by Liehr.2g
VI. The Localization Mechanism and Hz If electrons are localized in different parts of the molecule, exchange terms between these electrons are small. I n this ZocaZized state several different permutation states may be degenerate or near degenerate in zero order. Consequently, these states are highly mixed under the full Hamiltonian. (26) (a) G. Heraberg, Proc. Roy. Soc., A262, 291 (1961): (b) G. Heraberg and J. W. C. Johns, ibid., A295, 107 (1967); (c) G. Herzberg and J. Shoosmith, Nature (London) 183, 1801 (1959); (d) G. Herzberg, Can. J. Phys. 39, 1511 (1961); (26) (a) P. C. H. Jordan and H. C. Longuet-Higgins, Mol. Phye. 5, 121 (1962): (b) R.N.Dixon, Mol. Phys. 8,201 (1964). (27) See for example: (a) G. B,Porter and H. L. Schlafer, 2.Phye. Chem. (Frankfurt am Main), 37, 109 (1963); (b) H. L.Schlafer, H. Gausmann, and H. Zander, Inorg. Chern., 6, 1628 (1967); (c) J. Hempel and F. A. Matsen, J.Phys. Chem., 73,2502 (1969). (28) J. S. Griffith, “The Theory of Transition Metal Ions,” Cambridge University Press, 1964,p 306. (29) A.D.Liehr, J. Phys. Chem., 67, 1314 (1963). Volume 75, Number 8 August 1969
F. A. MATSENAND D. J. KLEIN
2484 Energy
% lT1
t
IOC
/
sc
/;il
reo*
L
HCH
Figure 6 . Two possible mechanisms for electronic relaxation of excited singlet methylene.
For diatomic molecule states in which the number of electron pairs in the separated atom limit is less than in the molecule, localization occurs for large internuclear separation. The extent of mixing of different permutational symmetries will always be large for sufficiently large R, since the exchange energy falls off exponentially” in R, whereas certain spin interactions fall offs1J2 as l/Ra and l/R4. As an example of a localizable system we consider the singlet and triplet states arising from the interaction of two ground-state hydrogen atoms. The exchange energy becomes small for large internuclear separations, so that the lowest singlet [SJ and triplet [TI states will be mixed by spin-interactions. Evaluating the total Hamiltonian including spin interactions over the zero-order antisymmetrical product kets, we find eigenkets of the form
IRK(S,I)(FwFb)(T);MF)
(6.1)
The spin X and the nuclear spin I quantum numbers are good for small R; the quantum numbers F a ($’a E 81 1,) and F b ( P b = g2 i b ) are good for very large R; MF is the only exact spin quantum number; r is a degeneracy index which is suppressed for otherwise unique kets. The amount of triplet character
+
+
%[TI E ~OO(RK(S,I)(FAJFB)(~); MF/e[T1 / K K ( ~ s (FA,FB)(r) ,I) ;MF) ( 6 . 2 ) as determined by Harriman and c o ’ ~ ~ o r k e is r splotted ,~~ in Figure 7. The Journal of Physical Chemistry
Figure 7. (Adapted from Harriman et al., ref 32) Per cent triplet character of the various states arising from the interaction of two 1s,ZS hydrogen atoms.
As the localization coordinate R increases, the exchange energy decreases and mixing increases. It is seen that Ro g loao appears to be a degeneracy bound for the localization range. Thus for R 2 Ro,the system is localized, and the permutational symmetries are mixed. Diatomic molecules a t large internuclear separations and in particular H2 provide simple examples of a biradical system.
VII. Biradicals A biradical state has been variously defined33as a triplet state, two noninteracting doublet states, or a state acting as though it has two monoradical functions. A state I K [ X ] K ’ [ X ’ ] ; Xwhich ) is localized, or nearly so, might be referred to as a biradical state. Biradical states either are of mixed permutational symmetry or are able to attain mixed permutational symmetry by a vibrational motion. As a consequence permutationally forbidden transitions in a biradical are allowed. (30) See, for example, C. Herring, Rev. Mod. Phya., 34, 631 (1962). (31) (a) Ur. J. Meath and J. Hirschfelder, J . Cliern. Phys., 44, 3197, 3210 (1965); (b) E. A. Power, W. J. Meath, and J. 0 . Hirschfelder, W1S.-TCI., 175 (1966); (c) W. J. Meath, J. Chem. Phys., 45, 4519 (1966). (32) J. E. Harrirnan, M. Twerdochilb, M . B. Milieur, and J. 0. Hirschfelder, Proc. Nat. Acad. Sci. U.S., 57, 1558 (1967). (33) (a) G . R. Freeman, Can. J . Chem., 44, 245 (1966); (b) L. N. Ferguson, “Electron Structures of Organic Molecules,” PrenticeHall Inc., Englewood Cliffs, N. J., 1952, p 211.
2485
SPIN-FREE QUANTUM CHEMISTRY Biradicals are frequently postulated to arise as intermediates in a numbers4 of chemical reactions and unimolecular isomerizations. Sometimes there are reasonable alternative concerted mechanisms in which the intermediate (or transition state complex) is not a biradical. Such a case of much interest36 involves the reactions of singlet [SI and triplet [TI methylenes with olefins. We note that whether or not a reaction is concerted is determined by the shape of the appropriate intermolecular potential surface,38 a dynamic property ; it is not determined by the permutational symmetry. If a chemical reaction passes through a biradical intermediate, intersystem crossover may occur because of the localization and consequent near degeneracy. An example is provided by the reaction of ground state aP carbon with an olefin double bond
Energy
I Coordinate
Rsoction
049.
,Q aaa.
PIP.
Figure 8. Possible potential for reaction 7.1. (The splitting of the perturbed curves I and I1 may be exaggerated in the region of the localized biradical intermediate.)
Energy
spiropentanes (71)
I
Experimental evidences7indicates that it is the biradical intermediate
which undergoes triplet to singlet intersystem crossing, rather than the intermediate
I
This is as we would expect, if the zero-order singlet and triplet states in the biradical intermediate are nearly degenerate. The process is indicated in Figure 8.
VIII. Photochemical Processes Intersystem crossover processes between singlet and triplet states are well known in photochemistry. Previous discussionsa8 of such processes have pointed out that a spin interaction must be responsible, usually3e the spin-orbit interaction. The role of Franck-Condon factors has been e m p h a s i ~ e d . The ~ ~ ~possible ~~ role of isomerization processes in radiationless processes has also been Here we wish to point out that near degeneracy of zero-order potential surfaces can enhance intersystem crossing processes42 in photochemistry. We assume that there is a vibrational coordinate Q for our molecule of interest, such that zero-order singlet [SI and triplet curves intersect as depicted in Figure 10. Thus the nonadiabatic intersystem crossing mechanism (4.16) may apply. We let the [SI and [TI curves of Figure 9 represent excited states of a molecule. A number of different photochemical processes are indicated in Figure 10. This figure is seen to be similar to
I
*Q
QzQo
Figure 9. Intersection of zero-order spin-free potential curves.
the usual Jablonski diagrams except that it includes the
Q E Qo near degeneracy states IK[S]K’[T];X) and (34) Recent reviews are: (a) P. D. Bartlett, Science, 159, 833 (1968); (b) W.A. Pryor, Chem. Eng. News, 46, No.3,70(1968). (35) (a) W.B. De More and 9. W. Benson, Advan. Photochem., 2 , 1 (1964); (b) P.P. Gaspar and G. S. Hammond in “Carbene Chemistry,” W. Kirmse, Ed., Academic Press, New York, N. Y., 1964, p 235. (36) R.Hoffman, J . timer. Chem. Soc., 90, 1475 (1968), (37) P.S.Skell and R. R. Engel, ibid., 88,3749 (1966), (38) (a) G. W. Robinson and R. P. Frosch, J. Chem. Phys,, 37, 1962 (1962); (b) G.W. Robinson and R. P. Frosoh, ibid., 38, 1187 (1963); (c) M. A. El Sayed, Accounts Chem. Res., 1, 1 (1968); (d) S. K. Lower and M. A. El Sayed, Chem. Rev., 66, 199 (1966); (e) R. B. Henry and M. Kasha, Ann, Rev. Phys. Chem., 19, 161 (1968); (f) M. Bixon and J. Jortner, J . Chem. Phys., 48, 715 (1968); ( 9 ) F.A. Matsen and D. J. Klein, Advan. Photochem., in press. (39) D. 8. McClure, J . Chem. Phys., 20,682 (1952). (40) (a) W.Biebrand, ibid., 47, 2411 (1967); (b) G. R. Hunt, E. F. McCoy, and I. G. Ross, Aust. J.Chem., 15,590 (1962). (41) D. Phillips, J. Lemaire, C. S. Burton, and W. A. Noyes, Jr., Advan. Photochem., 5,329 (1968). (42) (a) J. Franck and H.Sponer, J . Chem. Phys., 25, 172 (1956); (b) H.Sponer, RadiationRes. Suppl., 1, 658 (1959). volume 73, Number 8 August l g S 9
2486
li‘. A. MATSEN AND D. J. KLEIN
/K’[TI),s
Figure 10. Jablonski diagram showing the near degenerate IK[SjK’[T]; K > and IK(S)K’[T];K’> states at Q = Qo. Some primaTy photochemical processes are indicated: (i) absorption, (i’) fluorescence, (ii) and (ii’) intersystem crossing, (iii) and (iii’) vibrational relaxation and excitation, and (iv) phosphorescence.
] K[SIK’ [T];K’) and thus indicates an intersystem crossing mechanism. Process ii of Figure 10 may account for excited singlet to triplet intersystem crossing in benzene and other aromatic hydrocarbons. In the case of benzene it has previously been suggested43 that the observed rapid [SIto IT] intersystem crossing might involve states of mixed permutational symmetry, as for instance, I K [ S ] K ’ [ T ] ; Xand } IK[S]K’[T];X’). The nonadiabatic intersystem crossings (ii) and (ii’) could occur at a rate greater than for the case where there is no zero-order crossing oi potential curves. Such nonadiabatic intersystem crossing can be involved in delayed fluorescence. E-type delayed fluores~ e n c eis ~indicated ~ by processes (iii’), (ii’), and (i’). Another type of delayed fluorescence (ii), (ii’), and (i’) might occur when vibrational relaxation is slow.
IX. Conclusion The role of spin-free Hamiltonians and spin-free vector spaces in quantum chemistry has been reviewed. If the zero-order spin-free energy levels are widely separated and spin effects are small, molecular systems admit a spin-free formulation. The spin-free quantum number [A] was found to be a good approximate quantum number in such cases. Spin conservation was found to be equivalent to conservation of spin-free permutational symmetry. The spin-free analogs of the Wigner spin conservation rules were derived. The breakdown of local permutational symmetry, while conserving total spin-free symmetry, was discussed. Examples were given, with special attention to the case involving the effects of paramagnetic molecules in enhancing absorption in organic molecules. The breakdown of total spin-free permutational symmetry was found to occur when spin interactions in The Joumal of Phvsical Chemistry
the full Hamiltonian are taken into account. I n this case permutat,ional symmetry [A] and spin S are not good quantum numbers. The breakdown and consequent large mixing of states of different permutational symmetries was found to be especially acute near points of degeneracy. Such large mixing of permutational symmetries gave rise to two mechanisms, (4.13)and (4.16), for intersystem crossing. Methylene and Hz were discussed as examples where such intersystem crossings may play a role. It was found that biradicals could be formulated as localized states in which permutational symmetry is mixed or may easily become so. Intersystem crossovers involving states of mixed permutational symmetry were found to be possibilities in a number of photochemical processes.
Acknowledgments. The authors wish to acknowledge helpful discussions with Dr. A. A. Cantu and Dr. C. S. Burton.
Appendix. Permutational Symmetry Adaptation Symmetry adaption of a primitive space Itet ( K s F )
E VSF to the [Althirreducible representation of SN may be accomplished by the application of any element of a [X]th minimal invariant, subalgebra of 8,. Commonly used*~~,e945 elements of this algebra are a matric basis element e,,[’], an immanant projection operator e[’], or a structure projector .[’I. Symmetry adaptation of a primitive spin ket is accomplished in a similar manner. A primitive spin ket IM} E Vu of the form N/2+X
IM) 5
TI i= 1
N
a(i)
II
@(j)
(A. 1)
j=N / 2 + M+l
is an eigenket of k,, and thus kets projected from it will be also, This ket IM} has invariancea { y = ( N / 2 M , N / 2 - M I , and generates a space spanned by the kets PIM), P E SNu. This space is decomposed
1
+
The bases of these subspaces are
B(+]
;[xu]) = { / M ~ ; [ A ~ = ] ~1) tofiYi;[hul, , r = 1 tof[’ul)
(A,3)
We note
(43) G. B. Kistialrowsky and C. S, Parmeter, J. Chem. Phys., 42, 2942 (1965). (44) C.A. Parker, Advan. Photochem., 2,305 (1964).
SPIN-FREE QUANTUMCHEMISTRY
2487
and this last outer direct product frequency may be given by the spin-free conservation rules of eq 2.9 to yield the result p);[A"]
=
[A"] = [N 0, otherwise
{
3.9
- P, PI and N - 2P 2 2M (A. 5 )
Since frequencies only of 0 or 1 arise, we will suppress the degeneracy index 7 . We also note there is a matric basis such that
e,,[Aul(M) = ,a (Normalization)~M[A"]r) (A. 6) Space-spin lrets may be simultaneously symmetry adapted to SNsP8 SNuand S N s F I E ~E N " . Symmetry adaptation of the space-spin ket (KsF) 8 [ M ) to SNsF @ SNu is accomplished by symmetry adapting IKsF) to SNsF and lM) to SNC;symmetry adaptation to the [lN]-representation of SNsFH SNuis obtained on application of the antisymmetrizer a. Since the antisymmetrizer a restricts the SxsF8 8," symmetry N t o the form with [A"] = [XSF]], we will Iabel our symThus we metry-adapted fermion kets only with [ASF]. obtain the first part of (4.2)
[
[A'~]) = (P\Tormalization)a(lKSF)g~ /v IN [ASF]l.}) (A.7)
To obtain the second part of (4.2) we use2*
along with eq A.6
so that
l K S F [ r ] t@)
IKS"MIASF])= (Normalization) t
lM[X"]t) (A. 10) We briefly mention the difference between the bar and tilde symbols. Here [XI denotes the partition conjugate to [A]; the Young diagram for [XI has its rows and columns interchanged with that for [A]. Here the bar denotes more than conjugacy; it denotes a specific representation. The representation I' [I' is related to r [I' by ((-"
('-"
The representations and rIA1 may thus differ by a possible to choose unitary representation. It is always our representations such that Fix] = ,]'(?I except for the self-conjugate, [A] = [XI, irreducible representations. (46) R. D. Poshusta and R. W. Kmmling, Phys. Rev., 167 139 (1968).
Volume 75,Number 8 August 1966,
