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On nonadiabatic transition state theory
37, number
Volume
1
ON NONADIABATIC
1 lanuory
CHEMICAL PHYSICS LETTERS
TRANSlTION
1976
STATE THEORY It
Francis 3. McLAFFERTY and Thomas F. GEORCE~ Department ofChe~tistr_v, The EJrtiversity ofRocirester, Rochtesfer, New York 14627, US.4 Received 21 July 1975 Revised manuscript received 7-8 nupust
1975
A nonseparable semiclassical transition state approximation for reactions involving more than one eiectronic surface is rugestcd. The single surface formulation in terms of quasiprobabi!ity distributions used by Miller is discussed along with 3 sepxable semiclassical approximation for the nonadhbatic rate sugested in the Soviet literature. A thermally weraged nonadiabatic rate is defined, and a semiclassiclil appro&mation is presented, wherein the surface through which flux is calculated in the transition state approach is determ’aed by :Iie intersection of adiabatic electronic rurf~es viewed as functions of imaginary (or compteu) time.
1. Introduction
mates the Boltzmann
In the classical domain, transition state theory [l] is an excellent approximazion, in certain cases [2,3), to the exact thermally averaged birnoiecular rate constant for a single electronic surface. The essential physical idea incorporated into the theory is that one sits af an appropriate [I ,2] place and counts all positive ff us as reactive. This yieIds a variational bound to the exact rate constant. Due to the lack cf a well-behaved positive current operator, the quanium analogue of a variational bound to the exact quantum rate constant is not as useful [3,4]f. :n fact, any quantum transition state bound involving not only a smearing operation (associated with the definition of a positive current operator), but also the exac: nonlocal Boltzmann operator, will be inaccurate and requires the full solution of the corresponding scattering problem. Hence, the quantum anaiogue of transition state theory is not as viable as the classical theory. A semiclassical theory, which does not bound the rate constant and approxi*We FtefirUly acknowledge the U.S. Air Force Office of Scientific Research (Contact F43620-7-1X-0073) and the National Science Foundation for support of this research. ‘I CamiWmd Henry Dreyfus Fellow. 7 Tile bcund prksentecl in ref. 141 is more accurate in hi$cr
dimcnsjons sirlce it gutomatically includes the activation enagy &reaction, which is the doininvlt feature in.this
operator, should be more useful. Such a theory has been recently introduced by Miller [5]it, and it seems to generate a caIcuIabIe and ac-
curate representation cf sing!e (eiectrcnicj surface rate constants. In many reactions, mcce than one surface is involved, and a. generalization of transition state theory to these cases could be useful. Previous work /6-S] in this area has considered an approximate nonadiabatic transition state theory which involves a local separability approximation. From the experience of the singel curfact! case, we expect a nonseparable semiclassical theory to be the most useful form. The purpose of this paper will be to define 3 nonadiabatic average rate and extract a nonseparabie semiciassical approximation (ana!ogous to the single surface trea:ment [S]). The outline of the paper is as iollows. In section 2 we review the single surface formulation ES] from Moyal’s phase space poinr ofview [9,10] to establish notation and the appropriate method of approximation in tire two surface case. In section 3 we discuss an asymptotic (semiclassical) formulgtion of nonadiabatic transition state theory in the diabatic representation utilized by Soviet investigators [6-Sf . This formiilation, which is locally separable at the transition state, suo&ests the direction for section 4, where we derive a nonseparabte ft In the.notation.oP
ref. [41Lin one dime_nsion khiglcr in ref. [5] is proportional to Tr[JPiex:exp(-0H)PF;l *.vhcreQ = (t,~)-‘~~dklk) &I ad-[& is P free particte state.
:
_‘.
2i.7
: .,
:
.’
(.,
-..
‘,
1 January 1976
CHLNICAL PHYSICS LETTERS
semiclassic approximation to the nonadiabatic rate in the adiabatic representation. Section 5 is a discus. sibn.
2. Adhbhic
(singe
surface) transition
X e-~2A’/4ir2
e-i’/4A2
where G and j? are the position wd momentum operators and A is a transition state width [4], the associated quasiprobability distribution the;] becomes
state theory
.Restricting for simplicity our attention to one dimension, we can write the classical transition state rate constant for a potential barrier centered around the point .x = 0 as the proportionality [1,2]
(3rfz) -’ Tr [rii 1(x ’p)e-P’] = (2nfi)-l
Iplf1/2e-fifif1/2jp),
where Jp) isa free particle (1) where x and p are position and _tlomentum, respectively, II(P) is a step function and HC1 is the classical hamiltonian. The analogous quantum transition state result is
e -@’ = (3-ntl)-1SbrJdp,~Z(l.,,,)
(e-“‘)cl,
Tr[n^&x,&+]
where
.(e-“H),
(9
(3b)
mecharlical hamiltonian operator. Using the phase space representation of quantum mechanics we can espand_the quanrum positive current operator J^’ and e--Bff as (?+)cl>
state and
@a)
where 2 is the quantum
J^+ = (27itl)-+xj-dp:$(X;&
momentum
e-.y2/;12.
Eq. (8) represents the husimi distribution [ 12,133. The upper bound used in ref. [4] with a gaussian smeari!lg furlction can then be interpreted as that transition state theory which takes the an tinormal ordered form for the classical positive current, which is equivalent to averaging the &ssical pojitive current with 3 husimi quasiprobabiIity distribution*. Rather than treat the exact quantum positive current operator, we can consider a correspondence argument: we take as an approximation to the positive current operator the classical positive current (which need not be the same as the phase space function)
(2)
X-0: Tr[J^‘(x)emPH],
fcs) = (&-l/z
(8)
=pM$)S(x).
(10)
This defines a family of approximate quantum tion state theories de:ermined by the operator SiOJl Gl;I,(_~,p), = Tr[riil(x,p)e-O*
-1.
r^n 1(x,p) and
r;iz(x, p) are given sets of expansion ztors i9,l I] obeying Tr[liil(x,P)I;12(_T,P)]
= %rtls(~-x’)&b-p’),
(4b)
/i),2, = (3,~itl)--~ JcL$dp
oper-
(-5)
p/z @)6 (x)
-Ps]
X Tr[Gl(x,P)e If we choose have
trsnsiexpan-
(111
the Weyl expansion
operators
[9:10],
we
so that (2) becomes h: a (?r;fi)-‘J&J
dp (j’)cl(e-Pii)cl.
(6)
‘The interpretation of eqs. (3) and (6) is that we order J^+.in a ps;ticular fashion determined by “7, (XJI) and then average the “phase space” result with the quasiprobability,distribution associated with ~e~(x,p). For example, if we antinormal [ 111 orderd J + by writing A
. Iill (k, p). aS
:
k
111,v = ss dr
Q’p %‘)6
where the notation ’ 1t is interesting
W/,(-??‘)~
(13)
is the same as in eq. (7), and
to note thai within the Bargmann representatier. of a radiation field (14!, the husimi tiistribution. i.e., the “prdbability” ol observing the system in a coherent state, can be expressed as a phase vace path inte&.
Volume 37, number
1
CHIWLCXL PHYSKS
is the IV@er function associated wi‘rh e-PH. This is Miller’s choice [5]. In the semiclassical limit the Wig ner function becomes [S]
where the integrand is interpreted hamiltonian.
3. Thermally averaged nonadiabatic probabilities
as an effective
transition
Restricting ourselves for simplicity to collinear actions (A+BC-+ABtC), we can define a general
re-
thermally averaged rate constant for transitions between adiabatic electronic states i and j, as (for i=j there is no overall eler,troniz transition)
LETTERS
1 Jonuzry 1976
intersection of the two diabatic surfaces, pIz is the momentum normal to n, pL7, is a Landau-Zenerprobability factor,? is a t~vo-d~lens~onal momentum vector and His the ~lami~tonian for motion on surface l(2) before(after) the intersectidn. This type of fonnuIation is closest in spirit to ordinary single surface transition state theory approximations~ We would Iike a similar flux fornlulat~on in the adiabatic representation, since the semiclassical form for the S-matrix elements in eq. (16) is given in the adiabatic representation in terms of adiabatic surfaces WI(~) and Wz(q) (q is a twodimensional position vector). Initially there wo\;id seem to be some difficulty since surfaces (of the same syinmetrjr) do not intersect for any real vaiues of the coordinates [ 171’. However, we can define the line through which flux must be calculated by the real part of a complex intersection (e.g., the line Re s*(p) = 0 where s+ is a complex intersection point corresponding to a (real) value of the coordinate p perpendicular to s). In analogy to eq. (17) we can then write the I’ollowing approximate flux integral in the adiabatic representation:
Heft- is the sum of the nuclear kinetic energy and the effective surface fGn given as an S-matrix element for the transition from vibrational state “BC of SC on electronic surface i to vibrational state lzAB of AB on electronic sur-
%IAB,i%BCis
energy levels of face j, and gngC denotes vibrational BC on surface i. In analogy to the singIe surface investigations of George and hliller [ 151, we could make
the semiclassical approximation [16] to the S-matrix elements in eq. (I 6) artd perform the thermal average by’steepest descents. This, however, is not in the spirit of the ordinary transition state approximation. To suggest an appropriate direction in which to proceed, we first consider Soviet investigations which have been carried out in the diabatic representation f7,S], where an average, asymptotic (semiclassical) rate is defined as proportiona to 5 flux integral of the form (i=l , i=2j (171 The line (R) normal to the reaction coordinate s through which we caiculate the,ffux is defined by the
Ic;(IJ)~![-R~s”@)], (17) %K = W,(g)A[Res”(p)]+ and ,YJ~is a Stueckelberg-like probability factor of the form e-?-‘, wflere f is the imaginary classicat action associated with the transition from surface 1 to surface 2. The direction in nuclear momentum space along which this transition occurs can be chcsen normal to the localized Iir?e defined by the real part of the complex intersection. For the case where there is ;Lseam, this essentially reduces to the Tully--Preston [i8] ‘, model where the direction of transition is perpendicular to the seam. Eq. (18) defines a %dden” approximation [ 18,191 describing motion on surface I until we reach the transition state, whereupon motion CO!Itinues on surface ‘2. This equation, which makes a separable approximation locally at the transition state; sets the stage for the derivation of a nonseparable * TWO surfclccr, of n dimen;ions routd actually intersect in ;I real surface up to !I-:! dimensions, btit tkis surface is a subset ‘ora surface detined by the rcz1 pat of a complcs interscction.
; .‘..
G9
1 January
CHEMICAL PHYSICS LETTERS
Voiumr 37, number 1
path integral
semiclassical nonadiabatic rat.e, which is carried out in the next section with the hetp of a redxed Wigner function.
4, A semiclassical nonseparable tnnsilion state approximation
1976
(q21c217q’)
nonadiabatic
T4 is the nuclear kinetic energy operator and do? [q(P)] = QId[q@),r,r’ll2) for a given path &J), where d[q@),r:r’]
We can view eq. (16) BS an aarcrage flux quantum mechanically by considering
obeys the Bloch equation
[20]
P, + Gz(P)>41 d [q(P),f,f’! = 01
+
(26)
7’, is the electronic kinetic energy operator, V is the potential interaction and r represents electronic coordinates. If we expand d [q(p), r,r’] in terms of the adiabatic electronic wavefunctions Qi as
(21) is a reduced density operator, i;i a projector onto the final asymptotic electronic state and N a nuclear current operator. Fi is a projector onto all incoming solutions of the Schrddinger equation on surface i 2nd can be chosen as
d[4@);r,r’l
=Ca~Q~~i[q(P),rl~~[q(i7),r’l :
(22)
(27)
i.i
where where k is a rclztive wave number. the electronic and nuclear motion
In the limit that are separable, (20)
reduces to
=
(Za) where
(Xb)
rvik4t3)l9i[4tB)>Pl
WI
3
the initial
conditions
on the reacting
d&J@)]
=nT_ = 1:
system
are then
(29a)
42 [4(P)] = 0; = 0,
which is the sin& surface quantum mechanical result [S]: In a semiclassical theory cqrresponding to eq. (18) for two surfaces, we would expect the approximate phase space flus to be of the fo.m
and the equations -a+$
(296)
to be solved are
- f IV, a1 = Y1*L7,,
(30a)
-ao,lap-irrp, _ =T71al, where yyii ::’ $cLrd,&$~@. equations yield
(24)
(30b)
For only a single surface these
where Q&,12) is the l-educed /20] Wigner function, i.e., a Wigner transform over the nuclear degrees of free. dom for fiied electronic state, associated with the reduczd equilibrium densitv, mat$x:. [By equilibrium density mstriu we_mean Pe?){I; =_d”,rvheie [F, $ = 0. The&ore; ?kl/ap = -Hd XI~ d@=O) 5 p, SO that @ath integral expression exists and a reduced form
(31) diik(13)l = exp [ - j: d&Mil)l] 2 0 which, upon a Wigner transfcrm over nuclear degrees of freedom and symmetrization, gives ie the semiciassioal limit the single surface Wigner function [5,21]. For the two surface Wi
can be defined.]
This~matrix
hewe
?0’:.::
.,,.
car: be written
as the
kaginary.
(or complex)
time, can intersect.
,. .; :.
‘.,
.. :
.Tj,
.
”
:..
Us-
Volume
37, number L
ing the ordinary connection
CI-IEMLCAL PHYSICSLETTERS formulas
,r lailuary 1976
sion might be useful in the investigation of reactions with statistic4 behaviors.
we find
L i A ~micfassical statistical model has been discussed in rcfs. [19,7-21.
where 00 is an intekection point. Upon symmetrization, in analogy ti, the single surface treatment [S], we then find
References
J.C. Kcck, Advan. Chcm.‘Phys. 13 (1967) 85. P. Pechukas and F.J. hfclaffcrty, J. Chem. Phys. 58 (2973) 1622. N.S. Snider, unpublished. F.J. McLafferty arid P. Pechakas,
Chcm. Phys. Letters
27 (1974) 5 11. (33) where T is the classical nuclear kinetic. energy, and when eq. (33) is inserted into (24), WChave our final result.
5. Discussion We have suggested a generalization of transition state theory for a single potential energy surface. This involves the use of a reduced Wigner function representing the probability (in the semiclassical limit) of finding a reacting system at a given phase space point on surface 2 at “time” /3 if the system was on surface L initialy. A quantum calculation involving a projector, as defined in eq. (22) or (Bb), is difficult. However, as has been demonstrated for a sin$e surface, the semiclassical approxtination, where it is G&d,. would be an accurate and Iess tedious method of calculation. In the symmetrized form we would expect a calculation to be carried out forward in time from initial conditions along surface 2 and backward in time from initial conditions along surface 1. The difference in phase so accrued should generate the Stueckzlberg-like ps fact&s. in the genera1case of more than one transition region or cne transition region crossed many times, a possible extension of the average fh.~xapproximation wauld seem to invoIve urilizing connectioh formulas betlveen ‘“crossings” and c~cuIatin~ the flux ctssociated with the reduced Wigner function. Such an exten.
W.H. Miller, J. Chem. Phys. 61 (1974) 1823; 62 (1975) 1893; 63 (1975) 1166; S. Chapman, B.C. Garrett and W.H. Xlil!er, J. Chcnt. Phys. 63 (197.5) 2710. E.E. Nikitin. Advan. Quantum Chew 5 (1970) i35. Y.I. Khasats, A.K. Madumaron and h1.A. Vorotyntscv, Chcm. Sot. Faraday If 9 (1974) 1578. R.R. Dogonndzr and AX. Kuznetsov, Tear. i Ekspcrlm. Khim. 6 (1970) 298 [English transf. Theor. Enp. Chcm. 6 (1970) 7,433. F.J. McLafferty and T.F. George, J. Chem. Phys., submitted for pubfic3tion. J.R.K. Klauder and E.C.G. Sucfarshan, Fund~n~er~t~s of qunntum optics (Benjamin. New York, 1963). K.E. Cahill and R.J. Glauber, Phys. Rev. 177 (1969)
18.57. J. McKenna and H.L. Frisch, Phys. Rev. 145 (1966) 93: Xnn. Phys. 33 (1965) 136. I131 C.Y _ She 2nd H. Heffner, Phys. Rev. 152 (19&j 1103. J. Schweber, 1. Math. Phys. 3 11962) 831. T.F. George and W.H. ,\filler, J. Cficm. Plws. 57 (1972) 245 s. W.H. Xliller and T-F. George, J. Chem. Phys. 56 (I 972) 5637; T.F. George and Y.-W. Lin, J. Chem.~Phys. 60 (1974) 2340; F.J. hlcf.affercy snd T.F. George, J. Chcm. Pflys. 63 (1975) 1609. T.F. George, K. Morokuma and Y.-W. Lin, Chem. Pti)‘s.
Letters 30 (1975) 54. J.C. Tulf~;and K.K. Preston, 5. Chcm. Phys. 55 (!971) 562. G& Zahr, R.K. Preston and W.H. Xiilicr, J. Chem, Phys. 62 (1975) 1127.. JS. Dahler, I. Chem. Phys. 30 (1959) 1447. W.H. hlillor, J. Chem. Phyr. 55 (1971) 3146; 58 (1973) 1664; S.M. IIornstein and W.H. Miller, Chem. Phys. Lcitcrs 13 (19723 2%. [22] WM. bfilfcr, I. Chcm. phyr. 52 (191itl) 543.
Volume
1
ON NONADIABATIC
1 lanuory
CHEMICAL PHYSICS LETTERS
TRANSlTION
1976
STATE THEORY It
Francis 3. McLAFFERTY and Thomas F. GEORCE~ Department ofChe~tistr_v, The EJrtiversity ofRocirester, Rochtesfer, New York 14627, US.4 Received 21 July 1975 Revised manuscript received 7-8 nupust
1975
A nonseparable semiclassical transition state approximation for reactions involving more than one eiectronic surface is rugestcd. The single surface formulation in terms of quasiprobabi!ity distributions used by Miller is discussed along with 3 sepxable semiclassical approximation for the nonadhbatic rate sugested in the Soviet literature. A thermally weraged nonadiabatic rate is defined, and a semiclassiclil appro&mation is presented, wherein the surface through which flux is calculated in the transition state approach is determ’aed by :Iie intersection of adiabatic electronic rurf~es viewed as functions of imaginary (or compteu) time.
1. Introduction
mates the Boltzmann
In the classical domain, transition state theory [l] is an excellent approximazion, in certain cases [2,3), to the exact thermally averaged birnoiecular rate constant for a single electronic surface. The essential physical idea incorporated into the theory is that one sits af an appropriate [I ,2] place and counts all positive ff us as reactive. This yieIds a variational bound to the exact rate constant. Due to the lack cf a well-behaved positive current operator, the quanium analogue of a variational bound to the exact quantum rate constant is not as useful [3,4]f. :n fact, any quantum transition state bound involving not only a smearing operation (associated with the definition of a positive current operator), but also the exac: nonlocal Boltzmann operator, will be inaccurate and requires the full solution of the corresponding scattering problem. Hence, the quantum anaiogue of transition state theory is not as viable as the classical theory. A semiclassical theory, which does not bound the rate constant and approxi*We FtefirUly acknowledge the U.S. Air Force Office of Scientific Research (Contact F43620-7-1X-0073) and the National Science Foundation for support of this research. ‘I CamiWmd Henry Dreyfus Fellow. 7 Tile bcund prksentecl in ref. 141 is more accurate in hi$cr
dimcnsjons sirlce it gutomatically includes the activation enagy &reaction, which is the doininvlt feature in.this
operator, should be more useful. Such a theory has been recently introduced by Miller [5]it, and it seems to generate a caIcuIabIe and ac-
curate representation cf sing!e (eiectrcnicj surface rate constants. In many reactions, mcce than one surface is involved, and a. generalization of transition state theory to these cases could be useful. Previous work /6-S] in this area has considered an approximate nonadiabatic transition state theory which involves a local separability approximation. From the experience of the singel curfact! case, we expect a nonseparable semiclassical theory to be the most useful form. The purpose of this paper will be to define 3 nonadiabatic average rate and extract a nonseparabie semiciassical approximation (ana!ogous to the single surface trea:ment [S]). The outline of the paper is as iollows. In section 2 we review the single surface formulation ES] from Moyal’s phase space poinr ofview [9,10] to establish notation and the appropriate method of approximation in tire two surface case. In section 3 we discuss an asymptotic (semiclassical) formulgtion of nonadiabatic transition state theory in the diabatic representation utilized by Soviet investigators [6-Sf . This formiilation, which is locally separable at the transition state, suo&ests the direction for section 4, where we derive a nonseparabte ft In the.notation.oP
ref. [41Lin one dime_nsion khiglcr in ref. [5] is proportional to Tr[JPiex:exp(-0H)PF;l *.vhcreQ = (t,~)-‘~~dklk) &I ad-[& is P free particte state.
:
_‘.
2i.7
: .,
:
.’
(.,
-..
‘,
1 January 1976
CHLNICAL PHYSICS LETTERS
semiclassic approximation to the nonadiabatic rate in the adiabatic representation. Section 5 is a discus. sibn.
2. Adhbhic
(singe
surface) transition
X e-~2A’/4ir2
e-i’/4A2
where G and j? are the position wd momentum operators and A is a transition state width [4], the associated quasiprobability distribution the;] becomes
state theory
.Restricting for simplicity our attention to one dimension, we can write the classical transition state rate constant for a potential barrier centered around the point .x = 0 as the proportionality [1,2]
(3rfz) -’ Tr [rii 1(x ’p)e-P’] = (2nfi)-l
Iplf1/2e-fifif1/2jp),
where Jp) isa free particle (1) where x and p are position and _tlomentum, respectively, II(P) is a step function and HC1 is the classical hamiltonian. The analogous quantum transition state result is
e -@’ = (3-ntl)-1SbrJdp,~Z(l.,,,)
(e-“‘)cl,
Tr[n^&x,&+]
where
.(e-“H),
(9
(3b)
mecharlical hamiltonian operator. Using the phase space representation of quantum mechanics we can espand_the quanrum positive current operator J^’ and e--Bff as (?+)cl>
state and
@a)
where 2 is the quantum
J^+ = (27itl)-+xj-dp:$(X;&
momentum
e-.y2/;12.
Eq. (8) represents the husimi distribution [ 12,133. The upper bound used in ref. [4] with a gaussian smeari!lg furlction can then be interpreted as that transition state theory which takes the an tinormal ordered form for the classical positive current, which is equivalent to averaging the &ssical pojitive current with 3 husimi quasiprobabiIity distribution*. Rather than treat the exact quantum positive current operator, we can consider a correspondence argument: we take as an approximation to the positive current operator the classical positive current (which need not be the same as the phase space function)
(2)
X-0: Tr[J^‘(x)emPH],
fcs) = (&-l/z
(8)
=pM$)S(x).
(10)
This defines a family of approximate quantum tion state theories de:ermined by the operator SiOJl Gl;I,(_~,p), = Tr[riil(x,p)e-O*
-1.
r^n 1(x,p) and
r;iz(x, p) are given sets of expansion ztors i9,l I] obeying Tr[liil(x,P)I;12(_T,P)]
= %rtls(~-x’)&b-p’),
(4b)
/i),2, = (3,~itl)--~ JcL$dp
oper-
(-5)
p/z @)6 (x)
-Ps]
X Tr[Gl(x,P)e If we choose have
trsnsiexpan-
(111
the Weyl expansion
operators
[9:10],
we
so that (2) becomes h: a (?r;fi)-‘J&J
dp (j’)cl(e-Pii)cl.
(6)
‘The interpretation of eqs. (3) and (6) is that we order J^+.in a ps;ticular fashion determined by “7, (XJI) and then average the “phase space” result with the quasiprobability,distribution associated with ~e~(x,p). For example, if we antinormal [ 111 orderd J + by writing A
. Iill (k, p). aS
:
k
111,v = ss dr
Q’p %‘)6
where the notation ’ 1t is interesting
W/,(-??‘)~
(13)
is the same as in eq. (7), and
to note thai within the Bargmann representatier. of a radiation field (14!, the husimi tiistribution. i.e., the “prdbability” ol observing the system in a coherent state, can be expressed as a phase vace path inte&.
Volume 37, number
1
CHIWLCXL PHYSKS
is the IV@er function associated wi‘rh e-PH. This is Miller’s choice [5]. In the semiclassical limit the Wig ner function becomes [S]
where the integrand is interpreted hamiltonian.
3. Thermally averaged nonadiabatic probabilities
as an effective
transition
Restricting ourselves for simplicity to collinear actions (A+BC-+ABtC), we can define a general
re-
thermally averaged rate constant for transitions between adiabatic electronic states i and j, as (for i=j there is no overall eler,troniz transition)
LETTERS
1 Jonuzry 1976
intersection of the two diabatic surfaces, pIz is the momentum normal to n, pL7, is a Landau-Zenerprobability factor,? is a t~vo-d~lens~onal momentum vector and His the ~lami~tonian for motion on surface l(2) before(after) the intersectidn. This type of fonnuIation is closest in spirit to ordinary single surface transition state theory approximations~ We would Iike a similar flux fornlulat~on in the adiabatic representation, since the semiclassical form for the S-matrix elements in eq. (16) is given in the adiabatic representation in terms of adiabatic surfaces WI(~) and Wz(q) (q is a twodimensional position vector). Initially there wo\;id seem to be some difficulty since surfaces (of the same syinmetrjr) do not intersect for any real vaiues of the coordinates [ 171’. However, we can define the line through which flux must be calculated by the real part of a complex intersection (e.g., the line Re s*(p) = 0 where s+ is a complex intersection point corresponding to a (real) value of the coordinate p perpendicular to s). In analogy to eq. (17) we can then write the I’ollowing approximate flux integral in the adiabatic representation:
Heft- is the sum of the nuclear kinetic energy and the effective surface fGn given as an S-matrix element for the transition from vibrational state “BC of SC on electronic surface i to vibrational state lzAB of AB on electronic sur-
%IAB,i%BCis
energy levels of face j, and gngC denotes vibrational BC on surface i. In analogy to the singIe surface investigations of George and hliller [ 151, we could make
the semiclassical approximation [16] to the S-matrix elements in eq. (I 6) artd perform the thermal average by’steepest descents. This, however, is not in the spirit of the ordinary transition state approximation. To suggest an appropriate direction in which to proceed, we first consider Soviet investigations which have been carried out in the diabatic representation f7,S], where an average, asymptotic (semiclassical) rate is defined as proportiona to 5 flux integral of the form (i=l , i=2j (171 The line (R) normal to the reaction coordinate s through which we caiculate the,ffux is defined by the
Ic;(IJ)~![-R~s”@)], (17) %K = W,(g)A[Res”(p)]+ and ,YJ~is a Stueckelberg-like probability factor of the form e-?-‘, wflere f is the imaginary classicat action associated with the transition from surface 1 to surface 2. The direction in nuclear momentum space along which this transition occurs can be chcsen normal to the localized Iir?e defined by the real part of the complex intersection. For the case where there is ;Lseam, this essentially reduces to the Tully--Preston [i8] ‘, model where the direction of transition is perpendicular to the seam. Eq. (18) defines a %dden” approximation [ 18,191 describing motion on surface I until we reach the transition state, whereupon motion CO!Itinues on surface ‘2. This equation, which makes a separable approximation locally at the transition state; sets the stage for the derivation of a nonseparable * TWO surfclccr, of n dimen;ions routd actually intersect in ;I real surface up to !I-:! dimensions, btit tkis surface is a subset ‘ora surface detined by the rcz1 pat of a complcs interscction.
; .‘..
G9
1 January
CHEMICAL PHYSICS LETTERS
Voiumr 37, number 1
path integral
semiclassical nonadiabatic rat.e, which is carried out in the next section with the hetp of a redxed Wigner function.
4, A semiclassical nonseparable tnnsilion state approximation
1976
(q21c217q’)
nonadiabatic
T4 is the nuclear kinetic energy operator and do? [q(P)] = QId[q@),r,r’ll2) for a given path &J), where d[q@),r:r’]
We can view eq. (16) BS an aarcrage flux quantum mechanically by considering
obeys the Bloch equation
[20]
P, + Gz(P)>41 d [q(P),f,f’! = 01
+
(26)
7’, is the electronic kinetic energy operator, V is the potential interaction and r represents electronic coordinates. If we expand d [q(p), r,r’] in terms of the adiabatic electronic wavefunctions Qi as
(21) is a reduced density operator, i;i a projector onto the final asymptotic electronic state and N a nuclear current operator. Fi is a projector onto all incoming solutions of the Schrddinger equation on surface i 2nd can be chosen as
d[4@);r,r’l
=Ca~Q~~i[q(P),rl~~[q(i7),r’l :
(22)
(27)
i.i
where where k is a rclztive wave number. the electronic and nuclear motion
In the limit that are separable, (20)
reduces to
=
(Za) where
(Xb)
rvik4t3)l9i[4tB)>Pl
WI
3
the initial
conditions
on the reacting
d&J@)]
=nT_ = 1:
system
are then
(29a)
42 [4(P)] = 0; = 0,
which is the sin& surface quantum mechanical result [S]: In a semiclassical theory cqrresponding to eq. (18) for two surfaces, we would expect the approximate phase space flus to be of the fo.m
and the equations -a+$
(296)
to be solved are
- f IV, a1 = Y1*L7,,
(30a)
-ao,lap-irrp, _ =T71al, where yyii ::’ $cLrd,&$~@. equations yield
(24)
(30b)
For only a single surface these
where Q&,12) is the l-educed /20] Wigner function, i.e., a Wigner transform over the nuclear degrees of free. dom for fiied electronic state, associated with the reduczd equilibrium densitv, mat$x:. [By equilibrium density mstriu we_mean Pe?){I; =_d”,rvheie [F, $ = 0. The&ore; ?kl/ap = -Hd XI~ d@=O) 5 p, SO that @ath integral expression exists and a reduced form
(31) diik(13)l = exp [ - j: d&Mil)l] 2 0 which, upon a Wigner transfcrm over nuclear degrees of freedom and symmetrization, gives ie the semiciassioal limit the single surface Wigner function [5,21]. For the two surface Wi
can be defined.]
This~matrix
hewe
?0’:.::
.,,.
car: be written
as the
kaginary.
(or complex)
time, can intersect.
,. .; :.
‘.,
.. :
.Tj,
.
”
:..
Us-
Volume
37, number L
ing the ordinary connection
CI-IEMLCAL PHYSICSLETTERS formulas
,r lailuary 1976
sion might be useful in the investigation of reactions with statistic4 behaviors.
we find
L i A ~micfassical statistical model has been discussed in rcfs. [19,7-21.
where 00 is an intekection point. Upon symmetrization, in analogy ti, the single surface treatment [S], we then find
References
J.C. Kcck, Advan. Chcm.‘Phys. 13 (1967) 85. P. Pechukas and F.J. hfclaffcrty, J. Chem. Phys. 58 (2973) 1622. N.S. Snider, unpublished. F.J. McLafferty arid P. Pechakas,
Chcm. Phys. Letters
27 (1974) 5 11. (33) where T is the classical nuclear kinetic. energy, and when eq. (33) is inserted into (24), WChave our final result.
5. Discussion We have suggested a generalization of transition state theory for a single potential energy surface. This involves the use of a reduced Wigner function representing the probability (in the semiclassical limit) of finding a reacting system at a given phase space point on surface 2 at “time” /3 if the system was on surface L initialy. A quantum calculation involving a projector, as defined in eq. (22) or (Bb), is difficult. However, as has been demonstrated for a sin$e surface, the semiclassical approxtination, where it is G&d,. would be an accurate and Iess tedious method of calculation. In the symmetrized form we would expect a calculation to be carried out forward in time from initial conditions along surface 2 and backward in time from initial conditions along surface 1. The difference in phase so accrued should generate the Stueckzlberg-like ps fact&s. in the genera1case of more than one transition region or cne transition region crossed many times, a possible extension of the average fh.~xapproximation wauld seem to invoIve urilizing connectioh formulas betlveen ‘“crossings” and c~cuIatin~ the flux ctssociated with the reduced Wigner function. Such an exten.
W.H. Miller, J. Chem. Phys. 61 (1974) 1823; 62 (1975) 1893; 63 (1975) 1166; S. Chapman, B.C. Garrett and W.H. Xlil!er, J. Chcnt. Phys. 63 (197.5) 2710. E.E. Nikitin. Advan. Quantum Chew 5 (1970) i35. Y.I. Khasats, A.K. Madumaron and h1.A. Vorotyntscv, Chcm. Sot. Faraday If 9 (1974) 1578. R.R. Dogonndzr and AX. Kuznetsov, Tear. i Ekspcrlm. Khim. 6 (1970) 298 [English transf. Theor. Enp. Chcm. 6 (1970) 7,433. F.J. McLafferty and T.F. George, J. Chem. Phys., submitted for pubfic3tion. J.R.K. Klauder and E.C.G. Sucfarshan, Fund~n~er~t~s of qunntum optics (Benjamin. New York, 1963). K.E. Cahill and R.J. Glauber, Phys. Rev. 177 (1969)
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Letters 30 (1975) 54. J.C. Tulf~;and K.K. Preston, 5. Chcm. Phys. 55 (!971) 562. G& Zahr, R.K. Preston and W.H. Xiilicr, J. Chem, Phys. 62 (1975) 1127.. JS. Dahler, I. Chem. Phys. 30 (1959) 1447. W.H. hlillor, J. Chem. Phyr. 55 (1971) 3146; 58 (1973) 1664; S.M. IIornstein and W.H. Miller, Chem. Phys. Lcitcrs 13 (19723 2%. [22] WM. bfilfcr, I. Chcm. phyr. 52 (191itl) 543.