Random-Matrix Approach to Transition-State Theory

To model a complex system intrinsically separated by a barrier, we use two random Hamiltonians, coupled to each other either by a tunneling matrix element or by an intermediate transition state. We study that model in the universal limit of large matrix dimension. We calculate the average probability for transition from scattering channel coupled to the first Hamiltonian to a scattering channel coupled to the second Hamiltonian. Using only the assumption that the sum of transmission coefficients of channels coupled to the second Hamiltonian is large we retrieve transition-state theory in its general form. For tunneling through a very thick barrier independence of formation and decay of the tunneling process hold more generally.


I. INTRODUCTION
Barrier penetration and transition over a barrier play important roles in several areas of quantum physics.Since the pioneering work of Bohr and Wheeler [1], transition-state theory has been an important element in the theory of nuclear fission [2].It also plays a key role in physical chemistry [3].In two recent papers, Bertsch and Hagino [4] and Hagino and Bertsch [5] have studied a statistical model for a transition-state process wherein the complete mixing of states is hindered by an internal barrier.The model consists of two uncorrelated Hamiltonians, each a member of the Gaussian Orthogonal Ensemble (GOE) of random matrices.The first random Hamiltonian is fed from some entrance channel.Penetration through the barrier separating the two Hamiltonians is due to a single channel.We believe that for transitionstate theory, such a random-matrix approach is of considerable interest.Random-matrix theory is a universal tool for modeling complex quantum systems [6].It exhibits generic features of such systems without resorting to specific dynamical assumptions.These features emerge in the limit of infinite matrix dimension which allows for a clean separation of local fluctuation properties (that are generic) and global properties (that are not).
The present paper is motivated by the fact that the model used in Refs.[4,5] does not possess a realistic GOE limit of large matrix dimension N .In brief, in the propagators of the model Hamiltonians, the authors replace the energy E by E + iΓ.The constant offset Γ is supposed to account for the coupling to open channels.Inspection shows that the number of channels so simulated is equal to the matrix dimension N .If Γ is independent of N , the number of open channels tends to infinity with N .It is easily checked that the fluctuations worked out in Refs.[4,5] then tend to zero for N → ∞.That is unrealistic.The issue is presently addressed by the authors [7].We do not go into detail because we aim at presenting a viable alternative to the approach by * haw@mpi-hd.mpg.de Bertsch and Hagino.We investigate a model which does possess a realistic GOE limit with non-vanishing fluctuations.
Actually we study two simplified versions of the model of Refs.[4,5].Each employs two coupled GOE Hamiltonians.In the first, the two GOE Hamiltonians are coupled by a rank-one interaction.That is a model for barrier penetration via a single channel.In the second, the two GOE Hamiltonians are coupled via a single state.That models a physical transition state located right above the barrier.In each case, we calculate the probability for the transition from the entrance channel to some final channel on the other side of the barrier.We formulate the conditions under which that probability is the product of two or three statistically uncorrelated factors.Then the decay following barrier penetration or passage through the transition state is independent of its mode of formation.Such independence is the hallmark of transition-state theory.If there are sufficiently many open channels coupled sufficiently strongly to the second Hamiltonian, we retrieve the standard expression of transition-state theory.
The two versions of our model Hamiltonian are defined in Section II.The coupling to the channels and the scattering matrix are defined in Section III.In Section IV we define the conditions of validity of transition-state theory within our model and display the resulting transition probability.The special case of a thick barrier is treated in Section V. Section VI contains a brief summary.Technical details are deferred to an appendix.

II. TWO HAMILTONIANS
The tunneling Hamiltonian is The index T denotes the transpose, and V is a rankone interaction that models tunneling.The Hamiltonian which carries the transition state with energy E 0 is Both H 1 and H 2 are coupled to the transition state by vectors V 1 and V 2 , respectively.In Eqs.(1,2), H 1 and H 2 are GOE matrices of dimension N ≫ 1 each, defined in two Hilbert spaces denoted as space 1 and space 2, respectively.The elements of the matrices H 1 and H 2 are uncorrelated zero-centered Gaussian random variables with second moments (3) Angular brackets denote the ensemble average.Greek letters µ, µ ′ , µ 1 , µ ′ 1 , ρ, ρ ′ etc. range from 1 to N in space 1 while ν, ν ′ , ν 1 , ν ′ 1 , σ, σ ′ etc. range from N + 1 to 2N in space 2. The center row and column of the matrix (2) carry the index zero.The spectra of H 1 and H 2 range from −2λ to +2λ.For both Hamiltonians, the average level density as function of energy E is given by The rank-one interaction V in Eq. ( 1) is given by Here V is a constant with dimension energy, and (O 1 ) µN ((O 2 ) (N +1)ν ) are the elements of the N th column (of the first row, respectively) of two arbitrary N -dimensional orthogonal matrices in space 1 (space 2, respectively).Transforming and using the orthogonal invariance of the GOE leaves the ensembles H 1 and H 2 unchanged but changes V into For the Hamiltonian in Eq. ( 2), the vectors V 1 = {V 1µ } and V 2 = {V 2ν } are written, respectively, as Here {e 1µ } and {e 2ν } are unit vectors in N dimensions.Each of these forms one column of an orthogonal N -dimensional matrix O 1 and O 2 , respectively.Via a transformation similar to that in Eq. ( 6), the vectors V 1 and V 2 take the form Eqs. ( 1) and ( 7) on the one hand and Eqs. ( 2), ( 8) and ( 9) on the other define our two Hamiltonians.Without loss of generality we assume that V 1 , V 2 , V are positive.
In Section IV we show that to be physically meaningful, V must be independent of N and small compared to λ.An estimate for V 1 and V 2 is obtained by introducing the mean GOE level spacing d = πλ/N at the center We use the standard expressions for the spreading widths that account for the coupling of the transition state to the states in space 1 (space 2) with mean square coupling elements V 2 1 (V 2 2 , respectively), To be physically meaningful, Γ ↓ 1 and Γ ↓ 1 must be substantially smaller than the range of the GOE spectrum, and that same statement then applies to V 1 and V 2 .We define dimensionless positive parameters all of which are independent of N and small compared to unity.In what follows we think of the parameter Ṽ as being determined by a semiclassical calculation of the transition probability through the barrier.The parameters Ṽ1 and Ṽ2 measure the strength of the coupling of the transition state to spaces 1 and 2 and can only be determined by a fit to data.
When H 1 and H 2 each range over the orthogonal ensemble, the tunneling matrix element V becomes distributed uniformly over the states in space 1 and space 2. That follows from Eq. ( 5) as then the matrices O 1 and O 2 independently range over the set of orthogonal matrices in N dimensions.Therefore, tunneling is equally likely from any state µ in space 1 to any state ν in space 2. Pictorially speaking, in the model Hamiltonian (1) the tunneling barrier is replaced by a wall of uniform thickness that extends over the entire GOE spectrum.In an actual physical situation, the tunneling barrier has more or less the shape of an inverted parabola.Tunneling becomes ever more likely as the energy of the system approaches the top of the barrier from below.For the Hamiltonian (1) we simulate that feature by keeping the system's energy fixed at the center E = 0 of the GOE spectrum and increasing V.The top of the barrier is approached for very large V.The Hamiltonian (2) continues that physical picture to energies above the barrier where the transition process is enhanced by a resonance at energy E 0 .

III. SCATTERING MATRIX AND TRANSITION PROBABILITY
In the construction of the scattering matrix we follow Ref. [8].The open channels labeled a, a ′ , . . .(b, b ′ , . ..) are coupled, respectively, to states in space 1 (in space 2) by real coupling matrix elements W aµ (W bν ).Because of the normalization of the scattering wave functions, these matrix elements have dimension (energy) 1/2 .The number of channels coupled to either space 1 or space 2 is finite.It is held fixed as we take the limit N → ∞ of infinite matrix dimension in the following Sections.In spaces 1 and 2 we define the finite-rank width matrices Γ 1 and Γ 2 with elements For the Hamiltonians H tun and H tra we define, respectively, The elements for scattering from channel a to channel b of the scattering matrix S are with D given by Eqs. (14, 15) are completely general.In applications, however, the incident channel a consists of two fragments, each in its ground state.The final channels b consist of all pairs of fragments either in their ground or in excited states.In the converse reaction b → a, channel b consists of two fragments, each in its ground state.The final channels a consist of all pairs of fragments either in their ground or in excited states.We expand D −1 tun in powers of V.The resulting series is odd in V. We separate the first or the last factor V. The remaining series containing even powers of V can be resummed.We expand D −1 tra in powers of the elements in rows and labeled zero, respectively and proceed analogously.The resulting element S ab of the scattering matrix is the product of three dimensionless factors, with The dimensionless random variables ξ 1 , ξ 2 are defined as For the transition probability P ab from channel a to channel b, Eq. ( 16) gives For both Hamiltonians (1) and ( 2), the first (last) of the three factors gives the probability to enter (leave) the tunneling process or the transition state.The factor A tun in the first of Eqs.(17) accounts for tunneling, with the denominator accounting for repeated tunneling events, and correspondingly for the factor A tra in the second of Eqs.(17) which accounts for passage through the transition state.Each of the three factors in Eq. ( 19) is a random variable depending, in that order, on H 1 , on both H 1 and H 2 , and on H 2 .Therefore, the three factors are correlated.The probability of decay, given by the last factor, is not independent of the mode of formation of the tunneling process (or of the transition state).Averaging P ab over the two ensembles does not help because the average does not factorize.That shows that the standard assumption of transition-state theory does not hold in general.Transition-state theory holds if and only if the fluctuations of either ξ 1 or ξ 2 or both in the second factor in Eq. ( 19) are negligible in which case at least one of the three factors in Eq. ( 19) becomes uncorrelated with the rest.We now investigate the conditions for that to be the case.

IV. ASYMPTOTIC RESULTS
We determine the fluctuations of ξ 1 and ξ 2 by calculating mean values and variances 1 of these quantities in the limit of infite matrix dimension N .Eqs. (18) show that ξ 1 ↔ ξ 2 by simultaneously interchanging N ↔ N + 1, H 1 ↔ H 2 and Γ 1 ↔ Γ 2 .Therefore, we confine ourselves to ξ 1 .We use the close formal similarity of ξ 1 to the S-matrix describing scattering by the GOE Hamiltonian H 1 coupled to channels a, a ′′ .That matrix is defined by [8] S aa ′′ = δ aa ′′ (20) 1 We use the term variance for the expression |ξ − ξ | 2 .
The central piece of S aa ′′ is the propagator (E − H 1 + (i/2)Γ 1 ) −1 .That same propagator defines ξ 1 in the first of Eqs.(18).In Ref. [9] mean value and variance of S aa ′′ were calculated analytically for N → ∞ with the help of the supersymmetry approach.Adopting that calculation to the present case we obtain explicit expressions for average and variance of ξ 1 .Details are given in the Appendix.For the sake of completeness we mention that combining results of Refs.[10] and [11] yields an analytic expression for the full distribution of ξ 1 .The average, given by has magnitude unity and is independent of Γ 1 .The variance of ξ 1 does depend upon Γ 1 and is given by the threefold integral in Eq. ( 38) that involves the transmission coefficients These are defined in terms of the average diagonal elements of the GOE scattering matrix (20) and measure the strength of the coupling of channel a to space 1.For a single open channel with transmission coefficient T a the variance diverges like 1/T a for T a → 0. That is incontrast to the variance of S aa ′′ in Eq. ( 20) which is bounded from above by unitarity.The variance of ξ 1 decreases as the number of channels and the strength of their couplings to space 1 increase.If the sum of the transmission coefficients obeys a ′ T a ′ ≫ 1 we may use the asymptotic expansion of the threefold integral in inverse powers of a ′ T a ′ given in Ref. [12].As shown in the Appendix, the term of leading order is If the inequality in Eq. ( 23) applies, ξ 1 in Eqs.(17) may be replaced by ξ 1 , and transition-state theory applies.
In practice, the fluctuations of ξ 1 are sufficiently small if the number of open channels with transmission coefficients of order unity coupled to H 1 is of order 10 or bigger.
If the inequality in Eq. ( 23) holds, the first factor on the right-hand side of Eq. ( 19) is not correlated with the rest.It is then meaningful to calculate mean value and variance of the amplitude (the expression under the absolute sign).Using the same steps as for ξ 1 we show in the Appendix that the mean value of the amplitude vanishes and that for a ′ T a ′ ≫ 1 the variance, given by the leading term in the asymptotic expansion in inverse powers of a ′ T a ′ , is equal to Using Eqs. ( 22) and ( 24) and the corresponding results for ξ 2 , we can now give the asymptotic forms of the average transition probability.We distinguish three cases.
(i) The inequality b ′ T b ′ ≫ 1 holds, the variance of ξ 2 is small compared to unity but the variance of ξ 1 is not.Then We define Here A stands for either A tun or A tra defined in Eq. ( 25) as the case may be.Then The ensemble-averaged probability P a of formation of the transition channel or transition state is not accessible analytically and can only be calculated numerically.The normalized decay probability into channel b is given by T b / b ′ T b ′ and is independent of the mode of formation of the transition channel or transition state.
(ii) The inequality a ′ T a ′ ≫ 1 holds, the variance of ξ 1 is small compared to unity but the variance of ξ 2 is not.Then We define Here A stands for either A tun or A tra defined in Eq. ( 28) as the case may be.Then The ensemble-averaged decay probability P b in Eq. ( 29) is not available analytically and can only be calculated numerically.Therefore, Eq. ( 30) does not provide a useful parametrization of the reaction a → b.It does, however, usefully parametrize the converse reaction b → a defined below Eq. ( 15).For that reaction, the normalized probability to populate final channel a is given by T a / a ′ T a ′ , irrespective of the mode of formation of the transition channel or transition state.
(iii) The inequalities a ′ T a ′ ≫ 1 and b ′ T b ′ ≫ 1 both hold, the variances of ξ 1 and ξ 2 are both small compared to unity.Then and Here A stands for either A tun or A tra defined in Eq. ( 31) as the case may be.The normalized decay probability into channel b is given by T b / b ′ T b ′ and is independent of the mode of formation of the transition channel or transition state.The same statements apply to the converse reaction defined below Eq. ( 15).In contrast to Eqs. ( 27) and (30), Eq. ( 32) gives an explicit expression for the probability of both, the reaction a → b and the converse reaction b → a.

V. THICK BARRIER
A special case is tunneling through a thick barrier.Then Ṽ ≪ 1, suggesting that ξ 2 Ṽξ 1 Ṽ in the denominator of A tun in the first of Eqs.(17) may be neglected.For that to be the case, the square roots of the variances of ξ 1 and ξ 2 must be small in magnitude compared to 1/ Ṽ.Since 1/ Ṽ ≫ 1 that is a much weaker condition than imposed in Section IV where the square roots had to be small compared to unity.The condition is fulfilled already when only few channels are coupled to H 1 and to H 2 .In that case the asymptotic expressions in Section IV do not apply, and the variances of ξ 1 and ξ 2 must be obtained by direct calculation of the threefold integral in Eq. (38).If ξ 2 Ṽξ 1 Ṽ is indeed negligible, lowest-order perturbation theory in Ṽ is appropriate, does not fluctuate, P ab in Eq. ( 19) is the product of three statistically uncorrelated factors, and transitionstate theory holds.For P ab we find Formation and decay of the tunneling channel are independent processes.The associated probabilities, given by the ensemble averages in Eq. (34), can be calculated using the threefold integral (40) and its analogue for channel b.The formal symmetry of Eq. ( 34) with respect to the interchange a ↔ b reflects the symmetry of the scattering matrix in Eq. ( 14).Eq. (34) does not require that either a ′ T a ′ ≫ 1 or b ′ T b ′ ≫ 1 or both.It suffices that the square roots of both sums are substantially bigger than Ṽ, the (small) dimensionless tunneling matrix element through the barrier.That weaker constraint may be useful in some practical cases.Indeed, barrier penetration is small for energies far below the barrier.The number of open channels in either fragment decreases with decreasing excitation energy, reducing both a ′ T a ′ and b

VI. SUMMARY
To model a system intrinsically separated by a barrier, we have used two GOE Hamiltonians (each coupled to open channels) that are coupled to each other either by a tunneling matrix element, or by an intermediate transition state.We have studied that model in the universal limit of large matrix dimension of randommatrix theory.The transition probability P ab connecting an incoming channel a that feeds the first GOE Hamiltonian to an outgoing channel b coupled to the second Hamiltonian, is the product of three statistically correlated factors.These account, respectively, for formation of, passage through, and decay of the transition channel or transition state.
A sufficient condition for transition-state theory to hold in its standard form is that the sum of the transmission coefficients T b ′ accounting for decay of the second Hamiltonian into channels b ′ obeys b ′ T b ′ ≫ 1.Then the third of the above-mentioned factors becomes uncorrelated with the first two, the average transition probability factorizes, the decay of the transition channel or transition state is independent of its mode of formation, and the ensemble-averaged transition probability is The probability P a of formation of the transition chanor transition state is a parameter that is analytically available only if for the entrance channels a ′ the analogous condition a ′ T a ′ ≫ 1 holds as well.
Our formulation is symmetric with regard to the interchange a ↔ b.Therefore, the average probability for the converse reaction b → a is given by The conditions b ′ T b ′ ≫ 1 for the reaction a → b and a ′ T a ′ ≫ 1 for the converse reaction b → a are sufficient for the transition-state formulas (35) and (36), respectively, to hold.The weaker conditions ( a ′ T a ′ ) 1/2 ≫ Ṽ, ( b ′ T b ′ ) 1/2 ≫ Ṽ suffice for tunneling through a thick barrier with dimensionless tunneling matrix element Ṽ ≪ 1.Then the average probability for the reaction a → b factorizes.The probabilities for formation and decay of the tunneling channel given in Eq. (34) do not have the simple form T c / c ′ T c ′ but are available in terms of the threefold integral in Eq. (40).
Acknowledgement.The author is grateful to G. F. Bertsch for correspondence and to him and to K. Hagino for a clarifying discussion.
We turn to the amplitude of formation of the tunneling channel or transition state.The ensemble average of that amplitude vanishes for N → ∞.That is seen by following the steps that lead to Eq. (V.7.7).The calculation of the variance proceeds along the lines that lead to Eq. (38).It does not seem necessary to give the steps in detail.We find If a ′ T a ′ ≫ 1, we approximate the right-hand side of Eq. ( 40) by the first term of the asymptotic expansion in inverse powers of a ′ T a ′ .Eq. (4.8) of Ref. [12] then gives Eq. (24).Corresponding expressions hold for the last factor on the right-hand side of Eq. ( 19).