國立交通大學
應用化學系分子科學碩士班
碩士論文
一個更普遍用於半經典理論研究化學反應動態
學的絕熱勢能曲線
Constructing a More General Potential Curves for
Semi-Classical Method to Study State to State
Chemical Dynamics
研究生: 孔令鈞 (Ling-Ging Kaung)
指導教授: 朱超原 博士 (Dr. Chao-Yaun Zhu)
一個更普遍用於半經典理論研究化學反應動態
學的絕熱勢能曲線
研究生: 孔令鈞 指導教授: 朱超原
國立交通大學應用化學系分子科學碩士班
摘要
摘要
摘要
摘要
此研究利用了現今發展完善的反應途徑(Reaction coordinate)以及正交於反應 座標系的 3N-7 震動模(Normal modes)所建構的 J = 0 絕熱曲線(adiabatic curves)來
進行 OH + H2 H2O + H 化學反應動態學的研究。發現由此絕熱曲線在只考慮穿
隧效應下得到的 J=0 累積反應機率(Cumulative reaction probability)比 Miller 的 J=0 累積反應機率(Cumulative reaction probability)小很多。可能原因是反應端跟生成 端轉動相關的震動模的數目不相符。吾人也利用了絕熱模型(adiabatic model)，改 裝過的絕熱模型(modified adiabatic model)和透熱模型(diabatic model)來計算累積 反應機率，並得到三個可能與 Miller 結果差異的原因。第一是能量曲線並不是拋 物線的模型，第二是非絕熱偶合(non-adiabatic coupling)的貢獻和最後一個是延反 應途徑的頻率分析的準確度。為了確定哪個是最主要的原因，吾人利用上述的三
體系，並沒有一個精確的解。所以只有降維度的資料可以比較，也就是說，吾人 只能用降維度的絕熱模型來做比較。在與 Nyman 三個降維度；RLA，RLU 和 RBU, 的結果比較，吾人發現絕熱模型的結果與 Nynam 的結果的趨勢很近，並且發現 考慮的維度越小，共振的現象越大。從比較的結果，吾人發現以上三個原因中， 以第三個原因為決定累積反應機率準確度的最主要原因。
Constructing a More General Potential Curves for
Semi-Classical Method to Study State to State
Chemical Dynamics
Student: Ling-Ging Kaung Advisor: Dr. Chao-Yaun Zhu
M.S. Program, Institute of Molecular Science, Department of
Applied Chemistry, National Chiao Tung University
Abstract
We constructed more general adiabatic energy curves of J =0 for OH + H2
H2O + H by using the reaction coordinate and 3N-7 normal modes which is
orthogonal to the reaction coordinate, but found the cumulative reaction probability of
these adiabatic energy curves (without the effect of non-adiabatic coupling, only the
effect of tunneling) is lower than the result of Miller. The possible reason would be
the linkage between reactants and products. Since three rotational related vibration
modes in reactant side, but two in product side. And we also applied the adiabatic
model, modified adiabatic model and diabatic model to the cumulative reaction
result. First is that the energy curves are not parabolic model, the second one is that
the contribution of non-adiabatic coupling and the last one is the accuracies of the
frequencies along the reaction path. In order to make sure which one is the main
reason, we applied these three models to a larger but prototypical system, Cl +CH4
HCl + CH3. But for more than 5-atoms system, there’s no exact solution no matter
hyper-spherical (projected one is the same) or Jocabi coordinate. So only reduced
dimensions information is available, that is, only the adiabatic model with reduced
dimension could be applied to do comparisons. After comparing with the three models;
RLA, RLU, RBU, by Nyman, we found that the trends of our adiabatic models goes
well Nyman’s results and the effect of resonance becomes larger as the degree of
freedom becomes smaller. From the result, we conclude that the last reason; the
accuracies of the frequencies along the reaction coordinate dominate the accuracy of
謝誌
謝誌
謝誌
謝誌
在交大已經待了快六年半了，時間真是一轉眼就過了!這幾年經歷的各種快樂、 痛苦和挫折。這一路上很謝謝有你們。 首先我要感謝的是我的指導教授朱超原老師，感謝您在這一年半忍耐著我異想 天開的想法，並且給予我機會思考我做的每件事情，也很謝謝老師您對我耐心的 教導，讓我了解到理論的重要性。謝謝偉哥在我一開始進入實驗室，耐心的回答 我每一個很蠢的問題。謝謝俊吉學弟這個開心果，不論怎樣鬧你，你都嘻嘻哈哈 的。最後要感謝的是那些在我最痛苦的時候陪伴在我身邊的大學同學以及家人， 謝謝金毛、信良(後半年真的是謝謝你，要不是你陪我睡那燈超亮來以及蚊子多 到爆的實驗室，我大概會瘋掉八)、誠帥仔以及蝌蚪在這段時間總是靜靜聽我吐 苦水(雖然事後總是機車我，尤其是金毛)，更要謝謝我母親總是在我挫折的時候 鼓勵我，並且完全的相信我。謝謝你們給我這些美好的回憶。Table of Contents
Chapter 1 Introduction ... 1
1-1 Quasi-classical trajectory, quantum and semi-classical methods ... 2
1-1.1 Quasi-classical trajectory method ... 2
1-1.2 Quantum method ... 4 1-1.3 Semi-classical method ... 5 1-2 Motivation ... 5 References ... 9 Chapter 2 Theory ... 11 2-1 Born-Oppenheimer approximation ... 14 2-1.1 Hartree–Fock method (HF) ... 18 2-1.2 Post- HF methods ... 23
2-2 Adiabatic approximation for the motion of nuclei on single PES ... 27
2-2.1 Min energy path ... 28
2-2.2 Semi-classical theory ... 31
2-2.2.1 Zhu and Nakamura theory ... 31
2-3 Application to adiabatic curves on MEP ... 33
References: ... 44
Chapter 3 Results and Discussions ... 45
3-1 Theoretical results for OH+H2
H+H2O reaction ... 453-1.1 Theoretical calculation for MEP ... 45
3-1.2 Geometry of transition state ... 46
3-1.3 Normal modes along reaction path ... 46
3-2.Contruct the adiabatic curves of J = 0 on MEP ... 50
3-3. Cumulative reaction probability ... 51
3-3.1 Cumulative reaction probability for OH+H2
HCl+H2O ... 513-3.2.1 Theoretical results for the MEP of Cl + CH4
HCl + CH3... 59
3-3.2.2 Geometries of transition state, products and reactants ... 60 3-3.2.3 Normal modes along reaction path ... 60 3-3.2.4 Cumulative reaction probability for Cl + CH4
HCl +CH3... 61
Chapter4 Conclusions ... 99 Reference ... 101
Chapter 1 Introduction
One of the basic purposes of chemical reaction dynamics including state-to-state
reaction is to understanding reaction mechanism by analyzing the evolution
relationships between quantum states of products and reactants. These relationships
show the information of how bonds break and form gradually from reactant to product
through the transition state. Within Born-Oppenheimer approximation, the motion of
atoms in the molecule relies on the force created by the potential energy surface which
is formed by averaging motion of all electrons in the molecule.
A tri-atomic reaction, AB+CA+BC, is the most basic reaction for state-to-state
reaction dynamical studies. In this fundamental reaction involving only three
normal-mode vibrations plus rotation and translation, both experiments and theoretical
research have very much advanced in order to obtain the complete information for
diatom-atom reactions. These indicate that vibrational or translational excitation may
produce dramatic effects on chemical reactions. One of the most obvious effects is the
influence of chemical reactivity. 1 Polanyi has shown that the vibrational excitation is
more efficient than the translation excitation for increasing endoergic chemical
reactivity for atom-diatom-reaction, namely late barrier. The concept of early
barrier/late barrier becomes not so apparent for reactions containing polyatomic
there are a large number of vibrational modes. The quantum5-7 and quasiclassical
trajectory8 methods for atom-diatom reaction have been extended for studying
polyatomic reactions. At the same time, several exciting experimental results for
state-to-state reaction dynamics of polyatomic reactions have been found. Both these
experiments and theoretical research have pointed out that there are essential
differences between atom-diatom reactions and polyatomic reactions. Studying these
differences open new horizons for scientists.
1-1 Quasi-classical trajectory, quantum and semi-classical methods
Quasi-classical trajectory (QCT), quantum scattering and semi-classical methods
are three general methods to study chemical reaction dynamics theoretically. I would
like to introduce a brief review of QCT first, the second is quantum method, and the
third is semi-classical method. Finally, I will mention problems for each of three
methods.
1-1.1 Quasi-classical trajectory method
As mentioned above from Born-Oppenheimer approximation, the motion of
atoms in molecule is just experiencing force induced by PES formed. If potential
of the constituent atoms, it is possible to solve the motion of atoms on the surface for
collision trajectory in classical way. The initial conditions can be determined by using
coordinates and momentums of reactant molecule semi-classically, and then numerical
integration of the Hamilton equation of motion for coordinate Qj and it’s conjugate
momentum Pj=mdQj/dt are evaluated. Because of classical mechanical approach, there
are several quantum effects that have to be added separately:
1. Particles penetrating into the region are classical forbidden.
2. The zero point energy (ZPE) is absence. Several approaches9-15 have been done to
fix the problem of ZPE, but none of them are that reliable.
3. Classical mechanics does not recognize that reactants/products have quantum
states, the given integration results may produce the vibrational energy of products
different from quantum mechanics. But what we’re interested in is the quantum
mechanic features as the vibrational or rotational product distribution, in order to
obtain the correspondence, a binning process is needed. How to bin the region for
different vibration and rotation quantum number will influence the accuracy of
state to state dynamics, but this defect may cancel out for overall distribution.
4. Some of the initial-condition parameters of reactants would not be specified during
experiments, but the specification of these parameters is required in doing
the reactant vibration with respect to the time of collision, and so on. So, averaging
over these uncontrolled parameters of collision in the classical mechanic
calculation is required.
The defects of QCT can be easily seen from the mentioned above, the omission
of the quantum effects make it experienced difficulties at threshold; when the total
energy is slightly above the minimum required to cross the potential barrier, the ZPE
energy tends to be converted into translation energy, which is not allowed in quantum
mechanics. Besides this general problem for QCT, dealing with large system by QCT
is still a challenging problem. The reason for this is that determination of vibrational
actions; binning process, for large system is not a standard process. Even if there is
standard process for binning, obtaining the PES for large system is still difficult.
1-1.2 Quantum method
Apart from using the Newton mechanics to describe the motion of atoms; their
motion is governed by a wave equation in quantum mechanics. The general strategy is
to solve the time dependent or time independent Schrodinger equation in the
superposition wave function of channels interested and then obtain the quantities needed,
e.g. state to state cross section, by giving suitable boundary condition.
time. Since all accessible states; rotational and vibrational states, should be considered
in the calculation. Of course, one could calculate all the information, but are these
considerable quantities that demanding? Doing that requires huge effort, so a modified
method; reduced dimension quantum method is developed for this need. For four, five,
or six atoms system, this method is accessible. But for larger system, it’s a big problem.
So there’s reduced dimension quantum method (modified quantum method). However
for too large system, the modified quantum method is still not ok.
1-1.3 Semi-classical method
This method would be explained clearly in chapter 2. Same as the two methods
mentioned above. There is no general strategy for large system.
1-2 Motivation
As mentioned above, no matter quasi-classical trajectory, modified quantum
method or semi-classical method, there still exist difficulties for studying dynamics of
large system. So, this motivates us to produce a more general way; which is practical to
large system for dealing with state-to-state reaction dynamics. For system more than 5
atoms, it’s not that easy to obtain exact quantum results for reactions. In order to make
OH+H2H2O+H; which has been studied thoroughly from both theory and
experiments. Fristly, I would like to give a brief review of the calculations and
experiments which have been done for OH+H2H2O+H.
OH+H
2H
2O+H
The thermal rate constant for OH+H2; which is important in combustion
chemistry, has been measured over a wide range of temperature(T=250~2000K) for
both thermal distribution of reactants16-18 and there is obvious evidence18 that k(T)
shows non-Arrhenius behavior with a suggested best fit expression k(T) = 1.66
*10-16*T1.6*exp (-1660K/T) cm3/molecule*s. G.P. Glass19, Spencer et al 20 and R.
Zellner 18,21 investigate the influence of vibrational excitation on reaction rate.
They18,19-20 found that not small amount of enhancement (H2(v=1)/H2(v=0)>100) for
H2(v=1), but smaller effect(50% less than the excitation of H2) for OH(v=1); which is
reasonable since OH can be seen as a spectator in the reaction. Later, R. Zellner21
thought the large enhancement may be due to the possibilities of the contribution of a
translational rate acceleration of OH +H2 in the flow system. After correction, the
enhancement is rather small (5~66) compared to before (>100). Many fast-atom
experiments22-29 have been done for H+H2O(and it’s isotopomers) in the final
vibrational and rotational distribution of H2 and OH were determined.
Dunning’s calculation, and use it in transition state theory calculation for rate constant
with complex tunneling effect. Cohen et al31 used Isaacson’s reaction path potential in
transition state theory calculation of rate constant with new treatments of anharmoncity
of the transition state force field. A potential curve along a tunneling path is also
calculated by Truong and Evans32; which is used in calculating the rate constant.
Schatz and Elgersma33 based on the method for saddle point properties developed
by Walch and Dunning34 to construct a global potential energy surface. Extensive
quasiclassical trajectory calculations33,35,36 based on this PES have been done, they point
out interesting mode specificity in the H+H2O reaction in highly vibrational excited
states. There are several other QCT studies being reported later37-40
Quantum dynamics studies for the OH+H2 reaction have been reported by Wang
& Bownan41, Clary and coworkers42, Miller and coworkers43 , Zhang44, and Neuhauser45.
Those of Miller and coworkers , Zhang, and Neuhauser are full dimension. Wang &
Bownan RD-AB theory. Clary and coworkers use RBA-AB theory.
The cumulative reaction probability for J=0 has been calculated with full
dimension; by Miller, RD-AB; by Wang, and RBA-AB, by Clary. The results show that
there is a quite good agreement for RD-AB and RBA-AB with full dimension at low E,
but a little deviation at higher E.
approximation. The result for thermal rate constant along RD-AB (J-shift
approximation), RBA-AB (centrifugal-sudden approximation), and QCT are also
calculated. The RD-AB result is in good agreement with full dimension (RD-AB
results are reasonable, since J=0 CPR are almost the same and they both use J shift
method), but RBA-AB somewhat higher (The difference comes from using
centrifugal-sudden approximation). Finally, QCT’s result show good agreement with
full dimension at higher temperatures, but underestimate at lower temperatures (The
lack of tunneling effect for QCT method). The comparisons between full dimensions
show that taking bending motion as a adiabatic motion still give reliable result and
References
1.
J. C. Polanyi, Acc. Chem. Res. 5, 161 (1972).
2.
F. F. Crim, J. Phys. Chem. 100, 12725 (1996).
3.
R. N. Zare, Science 279, 1875 (1998)
4.
F. F. Crim, Acc. Chem. Res. 32, 877 (1999)
5. Bowman JM, ed. Advances in Molecular Vibrations and Collision
Dynamics: Quantum Reactive Scattering, 187 (1994)
6 .Clary DC, Echave 1. See reference 5
,2: 203(1994)
7.
Clary DC. J. Phys. Chem. 98, 10678 (1994)
8.
Schatz Ge. J. Phys. Chem. 99, 516(1995)
9.
Schatz GC. J. Chem. Phys. 79, 5386 (1983)
10. LuD-H, Hase WL. J. Chem. Phys. 89, 6723 (1988)
11.
Miller WH, Hase WL, Darling L. J. Chem. Phys. 91, 2863 (1989)
12 Bowman JM, Gazdy B, Sun Q. J. Chem. Phys. 91, 2859 (1989)
13. Sewell TD, Thompson DL, Gezelter JD, Miller WHo. Chem. Phys.
Lett. 193, 512 (1992)
14.
Peslherbe GL, Hase WL . J. Chem. Phys. 100, 1179 (1994)
15. Ben-Nun M , Levine RD. J. Chem. Phys. 101, 8768 (1994)
16. Tully FP, Ravishankara AR. J. Phys. Chem. 84, 3226 (1980)
17. Ravishankara AR, Nocovich JM, Thompson RL, Tully FP. J. Phys.
Chem. 85, 2498 (1981)
18. R. Zellner, J. Phys. Chem. 83, 18. (1979)
19. Glass OP, Chaturvedi BK. J. Chem. Phys. 75, 2749 (1981)
20.
J.E. Spencer, H. Endo and G.P. Glass, 16th Symposium (International)
on Combustion (The Combustion Institute, Pittsburgh) 829 (1976)
21. Zellner R, Steinert W. Chem. Phys. Lett. 81, 568 (1981)
22.
Kleinermanns K, Wolfrum J. Appl. Phys. B 34, 5 (1984)
23.Honda K, Takayanagi M, Nishiya T, Ohoyama H, Hanazaki I. Chem.
Phys. Lett. 180, 321 (1991)
24. Kessler K, Kleinermanns K. Chem. Phys. Lett. 190, 145 (1992)
25. Jacobs A, Volpp H-R, Wolfrum J. Chem. Phys. Lett.. 196, 249 (1992)
27. Jacobs A, Volpp H-R, Wolfrum J.J. Chem. Phys. 100, 1936 (1994)
28. Kippe S, Laurent T, Naik PD, Volpp H-R, Wolfrum J. Can. J. Chern.:
74, 615 (1994)
29. Adelman DE, Filseth SV, Zare RN. J. Chem. Phys. 98, 4636 (1993)
30. Isaacson AD. J. Phys. Chem. 96, 531(1992)
32.
Truong TN, Evans TJ. J. Phys. Chem. 98, 9558 (1994)
33. Schatz GC, Elgersma H. Chem. Phys. Lett. 73, 21(1980)
34.Walch SP, Dunning TH Jr. J. Chem. Phys. 72, 1303 (1980)
35.
Schatz Ge. J. Chem. Phys. 79, 5386 (1983)
37. Schatz OC, Colton MC, Grant JL. J. Phys. Chem. 88, 2971 (1984)
38.
Rashed O, Brown NJ. J. Chem. Phys. 82, 5506 (1985)
39. Harrison JA, Mayne HR. J. Chem. Phys. 87, 3698(1987)
40. Harrison JA, Mayne HR. J. Chem. Phys. 88, 7424(1988)
41.
Wang D, Bowman JM. J. Chem. Phys. 96, 8906 (1992)
42. Echave J, Clary DC. J. Chem. Phys. l00, 402 (1994)
43. Manthe U, Seideman T, Miller Who. J. Chem. Phys. 99, 10078 (1993)
44. Zhang DH, Zhang JZH. J. Chem. Phys. 99, 5615 (1993)
Chapter 2 Theory
As mentioned in chapter 1, state-to-state dynamics provide mechanism
understanding of reactions by indirect probing of transition state. What is probed are
the connections made between products and reactants, that is, these connections reveal
how bond break and form in the transition state. So the question is how to obtain the
information of state-to-state dynamics theoretically. In order to achieve this goal,
knowing how atoms move in the electron clouds of the molecule system is necessary.
In general, this kind of information would be gained by Born-Oppenheimer
approximation (or adiabatic approximation). Within Born-Oppenheimer approximation,
the motion of atoms in the molecule relies on the force created by the potential energy
surface which is formed by averaging motion of all electrons in the molecule, that is,
the motion of atoms and electrons could be considered separately as bellow
Nuc Elec
T =Ψ Ψ
Ψ (2.1) , where Ψ is the total wave function, T ΨElec is the wave function of electrons and
Nux
Ψ the wave function of nucleuses. Since the state-to-state dynamics here are talking about the states in the same electronic state, that is to say, only the
rovibrational states of the ground electronic state are necessary and non-adiabatic
coupling would occur between the rovibrational states. There are three general
However, no matter which method, there is still a big problem for dealing with large
system. It motivates us to generate a general way for polyatomic reaction. Our method
is based on the semi-classical theory developed by Zhu and Nakamura1, but chooses a
different reaction scattering axis for adiabatic curves, which is more general for
normal system (small or large system). Bellow I would like to talk about the general
concept for adiabatic approximation. Then, there will be explanation of the reason for
the chosen of this coordinate and the difference of semi-classical theory for dealing
with non-adiabatic coupling and its application.
General concept of adiabatic approximation
In a study of systems with many degrees of freedom, such as those consisting
several interacting particles, one general seeks to use the same way, a successive
reduction of multidimensional problem to several lower dimensional problems that
are simpler to deal with. There are two general way to do this, symmetry and
adiabatic separation. Adiabatic separation is main thought in Born-Oppenheimer
approximation, which is settled on the assumption that the motion associated with
some part of the variables (Fast) can be treated with other part (Slow) seen as frozen;
taking the energy of these fast variables at different slow variables produce effective
PES for the motion of slow variables. A general operator of this thought:
) , ( ) ( ) , (Qf Qs K Qs Had Qf Qs H = + (2.2)
, in which Qf and Qs represents the fast and slow variables, K(Qs) is the kinetic energy
for the motion in Qs which does not depends on Qf, and the adiabatic Hamiltonian
Had(Qf,Qs) is the operator of Qf which depends on Qs parametrically. The
schrondinger equation would be:
0 ) ( } ) , ( ) ( {K Qs +Had Qf Qs −E Ψ Qs = (2.3)
with total wave function to be: ) , ( ) ( ) (Qs =
∑
Fn Qsϕ
n Qf QS Ψ (2.4) , in which ϕn(Qf,QS) is from equation (2.5)) , ( ) ( ) , ( ) , ( f s n f S n S n f S ad Q Q Q Q E Q Q Q H ϕ = ϕ (2.5)
where E is the adiabatic energy and n
ϕ
n is the adiabatic state. Take equation (2.4) into equation (2.3) and integrate the coordinate of fast variables withϕ
m, you could obtain the non-adiabatic coupling terms as equation (2.6) and equation (2.7):) , ( ) , ( f S Q n f S m Q Q S φ Q Q φ ∇ (2.6) And ) , ( ) , ( 2 S f n Q S f m Q Q S φ Q Q φ ∇ (2.7) If the adiabaticity is pretty good, then the contribution of these two terms would be
pretty small, that is, we could omit these effects and obtain 0 ) ( } ) ( ) ( {K Qs +En Qs −E Fn Qs = (2.8) And the approximated total wave function as
) , ( ) ( ) ( s n S n f S n Q = F Q
ϕ
Q Q Ψ (2.9) From equation (2.2) to (2.7), there is no approximation inside until the function) , ( f S
n Q Q
ϕ is specified. Since it’s impossible to consider infinite basis in equation (2.5), we must do truncation. If the adiabaticity is good, then the non-adiabatic
coupling between adiabatic states; equation (2.6) and (2.7), would be small, that is,
the total wave function; equation (2.5), would be much more similar to the adiabatic
one, which means lesser basis are needed and this is where the approximation
originated. In order to obtain good approximation, the way of how to choose the fast
and slow variables is important, since the adiabaticity will be depended on how you
choose them. Born-Oppenheimer approximation is a good approximation of adiabatic
approximation. Bellow I would like to introduce this approximation.
2-1 Born-Oppenheimer approximation
Born-Oppenheimer approximation; named after Max Born and J. Robert
Oppenheimer, is a kind of adiabatic approximation which is used to describe the
motion of nuclei and electrons in the molecular system. As mentioned above, the
adiabaticity controls the accuracy of the approximation. Why adiabatic approximation
is a good approximation for the motion of nucleus and electrons in the molecular
slower than the motion of electrons, that is, we could see the movements of nuclei as
frozen when electrons are moving at every instant, which is pretty reasonable, since
the weight of nucleus is much larger than the weight of electron (even the lightest one;
hydrogen, are larger than electron in a factor of 1836). So it’s obvious to take the
motion of nuclei as slow variables and the motion of electrons as fast variables. From
the view of derivation, the nonrealistic Hamiltonian of the molecular system could be
written as
∑ ∑
∑∑
∑∑
∑
∑
> > + + − ∇ − ∇ − = A B A AB B A i j i ij A i Ai A i i e A A A r e Z Z r e r e Z m m H 0 2 0 2 0 2 2 2 2 2 4 4 4 2 2 ˆ πε πε πε h h (2.10), where A, B refer to nuclie and i, j refer to electrons. The Hamiltonian may be written
explicitly as ) ( ) ( ) , ( ) ( ˆ ) ( ˆ ˆ T R T r V R r V r V R H = N + e + Ne + ee + NN (2.11) , where R is the set of nuclear coordinates, r is the set of electronic coordinates. In
equation (2.11), it is VNe this term that makes the separation of electrons and nuclei
becoming impossible. But as mentioned above, since the motion of nuclei is much
slower than the motion of electrons, it is reasonable to rewrite the Hamiltonian as
) , ( ˆ ) ( ˆ ˆ T R H R r H = N + e (2.12) WhereHˆe(R,r)is as bellow
∑ ∑
∑∑
∑∑
∑
> > + + − ∇ = A B A AB B A i j i ij A i Ai A i i e e r e Z Z r e r e Z m r R H 0 2 0 2 0 2 2 2 4 4 4 2 ) , ( ˆ πε πε πε h (2.13)Or ) ( ) ( ) , ( ) ( ˆ ) , ( ˆ R r T r V R r V r V R He = e + Ne + ee + NN (2.14) And the total schrondinger equation of the molecular system would be
0 ) ) , ( ˆ ) ( ˆ ( ) ˆ (H−E Ψt = TN R +He R r −E Ψt = (2.15) , where He is the Hamiltonian of electrons at fixed nuclei configuration, R is the set of
nuclear coordinates; the slow variables, and r is the set of electronic coordinates: the
fast variables. The total wave function would be
∑
= Ψ n e n Nuc n t(R) F (R)ϕ
(R,r) (2.16) , where ϕne(R,r)is from ) , ( ) ( ) , ( ) , (R r R r E R R r He ϕne = n ϕne (2.17) , in which En is the adiabatic electronic energy and ϕne(R,r) is the adiabaticelectronic state. Take equation (2.16) back to equation (2.15) and integrate the coordinates of electrons withϕme(R,r), then we get
0 ˆ + − =
∑
Nuc m Nuc m m n r e n Nuc n N e m T F ϕ E F EF ϕ (2.18)And the term
∑
n r e n Nuc n N e m T F ϕ
ϕ ˆ could be further derived by
∑
∇ − = A A A N m T 2 2 2 h (2.19) Then we obtain[
2]
0 2 2 2 2 2 2 = ∇ ∇ + ∇ − − + ∇ −∑∑
∑
n A Nuc n A r e n A e m Nuc n r e n A e m A Nuc m m A A A F F m F E E m ϕ ϕ ϕ ϕ h h (2.20)A more compact notation is needed for equation (2.20), so we introduced the
following quantities used by Tully2
r e n A e m A mn R d( )( )= ϕ ∇ ϕ (2.21) r e n A e m A mn R D( )( )= ϕ ∇2ϕ (2.22)
These two terms are the non-adiabatic terms shown in equation (2.6) and (2.7). The
neglect of these two terms is the Born-Oppenheimer approximation, which is
reasonable since the weight of nucleus is much larger than the weight of electron, that
is, e
n
Aϕ
∇ is much smaller than Nuc
n AF ∇ . So we could obtain 0 2 2 2 = − + ∇ −
∑
m A m A A F E E m h (2.23)And total wave function would be
) , ( ) ( ) (R FnNuc R ne R r t =
ϕ
Ψ (2.24)Equation (2.17) and (2.23) are the two basic equations of Born-Oppenheimer
approximation, where Em is obtained by equation (2.17); the adiabatic electronic
energy, which is seen as the average field of the motion of nuclei. This average field is
calculated before dealing with the motion of nuclei, that is, solve the Shcrondinger
equation at every fixed nuclei configuration. Now the question comes to how to
obtain the adiabatic electronic energy state and energy, that is how to solve equation
(2.17). In general, there are two general methods for quantum calculation;
the Post–Hartree–Fock are the set of methods developed to improve on the
Hartree–Fock (HF), which considered the electron correlation energy that HF method
didn’t consider. Bellow we would to introduce the HF method and then some Post-HF
methods.
2-1.1 Hartree–Fock method (HF)
Hartree–Fock theory is one of the simplest approximate theories for solving
the many-body Hamiltonian, which reduce the many-body problem into several
one-body problems. It is based on the independent particle models, that is, the total
wave function of the electrons are the product of wave function of each single
electrons, as bellow ) , ( ) , ( ) , ( 1 2 2 1 Ne Ne e a e a e a e t =ϕ R r ϕ R r Lϕ R r Ψ (2.25)
Where R is the set of nuclear coordinates, ri is the ith electronic coordinate and ai is
the ith spin orbital function. In order to contain the property of anti-symmetry for the
total wave function of electrons, single Slater determinant is used to represent the total
wave function, as the form bellow
) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ! 1 2 1 2 1 2 1 2 2 2 1 1 1 e e N Ne Ne e e N e a e a e a N e a e a e a N e a e a e a e e SD r r r r r r r r r N
ϕ
ϕ
ϕ
ϕ
ϕ
ϕ
ϕ
ϕ
ϕ
L M L M M L L = Ψ (2.26) WithB A B A j j a a e a j e a *(r )ϕ (r )dr δ ϕ =
∫
(2.27) That is∫
Ψ *Ψ 1 2 =1 e N e SD e SD drdr Ldr (2.28). So the energy would be
e SD e e SD H E = Ψ ˆ Ψ (2.29)
WhereHˆ is the form as equation (2.13) or (2.14) which could be rewritten as e
∑ ∑
∑∑
∑
> > + + = M A M A B AB B A N i N i j ij N i i e r Z Z r H r R H e e e 1 ) , ( ˆ 0 (2.30) Where 0 iH is the operator of single electron at the same nuclei configuration as
bellow
∑
− ∇ = A Ai A i e i r e Z m H 0 2 2 2 0 4 2πε
h (2.31)If Ne equals to 2N, and the form of Slater determinant changed to
) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ! 1 2 2 1 2 2 1 2 2 1 1 1 1 1 1 1 N e B e B e B N e B e B e B N e B e B e B e e SD r r r r r r r r r N N N N
β
ϕ
β
ϕ
β
ϕ
β
ϕ
β
ϕ
β
ϕ
α
ϕ
α
ϕ
α
ϕ
L M L M M L L = Ψ (2.32)Then equation (2.29) can be further written as
∑∑
∑
+ − = N i N j B B B B N j Bj J i j K i j I E 2 (2 ) (2.33) Where the third term of equation (2.30) is not considered, since it is just an effect of aconstant and shifts to the eigenvalues and
i j j j i r e B i i e B B r H r I =
ϕ
( ) 0ϕ
( ) (2.34)m l j i j i j i m rr e B l e B lm m e B l e B B B r r r r r J = ϕ ( )ϕ ( ) 1 ϕ ( )ϕ ( ) (2.35) m l i j j i j i l rr e B m e B lm m e B l e B B B r r r r r K = ϕ ( )ϕ ( ) 1 ϕ ( )ϕ ( ) (2.36) j iB B
J is called coulomb integral and the
j iB B
K is called the exchange integral if i is not equals to j. By applying the vibration method to equation (2.34), we could find that
the form of spatial orbital will satisfy the following equation (2.37) in order to
minimize the energy; which is the famous Fock equation
N i r r r F j Be j B Be j i i i( ) ( ) 1 ,2 , , ) ( ˆ ϕ =ε ϕ = L (2.37)
WhereFˆ(rj)is the Fock operator, which is given by
[
]
∑
− + = N k j B j B j j f r J r K r r F k k ( ) ˆ ) ( ˆ 2 ) ( ˆ ) ( ˆ (2.38) Where∑
− ∇ − = A jA A j j r Z r f 2 1 ) ( ˆ (2.39) ) ( ˆ j B r Jk is the coulomb operator given by
m K K i i k m r e B mj m e B j e B j e B j B r r r r r r Jˆ ( )
ϕ
( )=ϕ
( )ϕ
( ) 1ϕ
( ) (2.40) And KˆB (rj)k is the exchange operator given by
m i K K i k m r e B mj m e B j e B j e B j B r r r r r r Kˆ ( )
ϕ
( )=ϕ
( )ϕ
( ) 1ϕ
( ) (2.41) So the expression for the ith molecular orbital would bej i i i j r e B j j e B B
ϕ
(r ) Fˆ(r )ϕ
(r )ε
= (2.42) Or∑
− + = N j B B B B B Bi I j (2J i j K i j)ε
(2.43)Comparing with equation (2.33), we could get the total energy in the form of
∑
+ = N i B Bj i I E ε (2.44) ,which is not just the summation of the orbital energy, since the interaction betweenelectrons will give contribution to the total energy. As it comes to solve the equation
(2.37), we found that equation (2.40) and (2.41) cannot be evaluated until all the
orbital are known. So self-consistent procedure is needed, in which you guess a set of
N coupled basis. Using the Fock equation, we could get a set of new orbital. Then this
new set of orbital are used to calculate the new Fock equation. Repeating this process
until the new set of orbital is almost same as the previous set, in other words, until
they are self-consistent. Now the problem comes to how to guess the orbital. This
problem is pretty important, since it controls the convergence time for the
self-consistent and the accuracy of the convergent result. The most general procedure
is developed by Clemens Roothaan. He expressed the molecular orbital e MO
ϕ as the linear combinations of basis functions
φ
v, which in general is atomic orbital∑
= k v v v e MO c φ ϕ (2.45) Take equation (2.45) back into equation (2.37) and integrate the electron coordinate rjwith
φ
µ*(rj). Then we can getk c S c F k v v v k v v v =
∑
=1 ,2 ,L,∑
µ ε µ µ (2.46) WhereFµvis given byj r j v j j v r F r r Fµ φµ( ) ( )φ ( ) ) = (2.47) andSµvis given by j r j v j v r r Sµ = φµ( )φ ( ) (2.48) If
φ
vis chosen a set of real functions; which is general the case. Then bothFµvandSµvare k x k Hermitian matrices. The equation (2.46) could be rewritten in matrix
notation as bellow
Sc
Fc=
ε
(2.49) Which is the most general form used in the computational calculation for HF method,because of the convenience of matrix notation. Up to now, we have shown the
derivation of HF method. From this, we could see that there is a principle for making
sure the accuracy of the HF ground state result; the lower the ground state energy the
more accurate the ground state result, since it is derived from variation method.
Besides that advantage, from equation (2.37) we found that the best form after
vibration method is one electron schrondinger equation with the effect of taking other
electrons as average field, that is, this not only reduce the multi-electrons problem
into several one-electron problems but also improve the molecular orbital; the Fock
orbital, which has the information of considered electron interacting with other
electrons and it is much better than normal single electron orbital. Although there are
particle model approximation for the total wave function, which is absolutely not the
form of exact wave function. Because of the second term in the right side of equation
(2.30), they are no longer independent. So it is impossible for the total wave function
in the form of the product by each single electrons wave function. In other words, the
result of HF for ground state will never same as the exact one and it will always
higher than the exact one (Since it is derived from variation method.), even if we used
infinite basis equation (2.45); which we call this HF limit. The difference between HF
limit and the exact one is in the order that will influence the accuracy of chemical
reaction. Since HF omit the effect of the instant interaction of electrons (because of
the form for total wave function); which is called electron correlation energy, and this
is very important for chemical reactions. In order to consider the effect of correlation
energy, there is the development of the post-HF method, which would be introduced
bellow.
2-1.2 Post- HF methods
As mentioned above, Post- HF method is the method improved to consider the
correlation energy that HF didn’t consider, which could be defined in the form of
bellow equation mit HF exact corr E E E = − lim (2.50)
There are several Post- HF methods. Here we would only mention the MP2 for
Moller-Plesset Perturbation theory and the CCSD, CCSD(T) for couple cluster
method.
Moller-Plesset Perturbation Theory
The basis of Moller-Plesset is to take the Slater determinants constructed by
Fock orbitals as the zero order function and further improve the energy and wave
function by perturbation theory. So the zero order schrondinger equation would be as
bellow ) 0 ( ) 0 ( ) 0 ( ) 0 ( ˆ i i i E H Φ = Φ (2.51) Where
∑
= Ne i i r F Hˆ(0) ( ) (2.52) In whichF(ri)is the Fock operator∑
= i a i i E(0)ε
(2.53) and (0) iΦ is constructed by Fock orbital(The excited state is simply obtained by
exciting the electrons to virtual Fock orbital and then construct its correspond Slater
determinants.) Since Hˆ is in the form of equation (2.30) andHˆ(0)is in the form of
equation (2.52), then we could expected the first order Hamiltonian would be
) 0 ( ) 1 ( ˆ H H H ) ) − = (2.54) Or
∑∑
∑∑
− − = e j j j e e N i a a a N i N j ij J i K i g H ( ) ˆ () 2 1 ˆ(1) ) (2.55) Where ij ij r g = 1 (2.56) From the basic of perturbation we would find that the energy considered to first order(MP1) is the result of HF, which is pretty reasonable since
1 ) 0 ( ) 0 ( ) 1 ( ) 0 ( MP i HF i HF i HF i i i i i i E E H H E E E = + = Φ Φ = Φ Φ = = ) ) (2.57)
So considering to second order energy (MP2), the energy would be
∑
∑∑
∑∑
≠ − Φ Φ Φ Φ − = + + = 0 ) 0 ( ) 0 ( ) 0 ( ) 0 ( ) 0 ( ) 0 ( ) 2 ( ) 1 ( ) 0 ( 2 1 2 1 A A i i N i N j ij A A N i N j ij i HF i i i i i E E g g E E E E E e e e e (2.58)From equation (2.58), it is easy to see that the second order energy correction must be
negative. So if we are talking about the ground state, and the basis already make HF
limit, then it’s obvious that the second order energy correction would be the electron
correlation energy.
Coupled-Cluster Theory
The concept of coupled-cluster is little bit same as configuration interaction, but
sort the groups in the order of number of electrons excited (same as MP2), where CI
sort them in the order of spin (converge too slow.). So total wave function would in
L L + Φ + Φ + Φ + Φ = Ψ
∑
∑
∑
abc ijk abc ijk abc ijk ab ij ab ij ab ij a i a i a i CI a a a , , , 0 (2.59)WhereΦ0is the HF ground wave function, i; j; k is the occupied orbitals and a; b; c is the virtual orbitals. The coefficients in equation (2.59) are obtained by variation
method, but not the case for coupled-cluster method, which are obtained by second
excitation as bellow, where the coefficients are not obtained by variation method
∑∑
Φ = Φ occ i vir a a i a i t T1 0 ) (2.60)∑∑
< < Φ = Φ occ j i vir b a ab ij ab ij t T2 0 ) (2.61)∑ ∑
< < < < Φ = Φ occ k j i vir c b a abc ijk abc ijk t T3 0 ) (2.62) So equation (2.59) would be(
1+ ˆ1+ ˆ2 + ˆ3 +)
Φ0 = Ψ T T T LL CC (2.63)By non linear transformation we could get
( ) ( ) 0 ˆ 0 ˆ ˆ ˆ 1 1 2 3 Φ = Φ = Ψ +T+T+T+ T CC e e L L (2.64)
, which is the exact solution to the equation (2.17). By multiplyinge−( )Tˆ and integrate withΦα, which is at least single excitation, then we can get the information of coefficient in it as bellow 0 ˆ 0 0 ˆ ˆ = Φ Φ = Φ Φ − α α e T T e H e (2.65)
By multiplyinge−( )Tˆ and integrate withΦ0, then we can get the total energy as bellow
0 2 1 2 0 0 ˆ ˆ 0 ˆ ) 2 1 ˆ ( ˆ ˆ Φ = = + Φ + Φ Φ − T T H E E e H e T e T HF e (2.66)
( )T e ˆ as bellow ( ) 2 1 ˆ ˆ ˆ 1 T T eT ≈ + + (2.67) And CCSD(T) is same as CCSD but consider some effect of triple excitation, but not
entirely as CCSDT.
2-2 Adiabatic approximation for the motion of nuclei on single PES
Now the question is how to deal with the nuclei motion on the single adiabaticelectronic PES. Adiabatic approximation is the general strategy for this kind of
problem. But electronic transition, vibrational transition seems cannot be a good
candidate for good adiabaticity. But in fact, reactive transitions occur in the
rovibraiontal states in physical different regions, reactants and products and occur
efficiently only in the region that two parts come closer. So using adiabatic separation
for state to state dynamics problem seems reasonable. Still, the adiabaticity is not that
good as electronic transition, that is, non-adiabatic transition still play an important
rule inside the state to state dynamics. But the adiabaticity is good enough that he
region of strong non-adiabatic coupling would be separated from each other, each of
them can be seen freely from others. Because of the properties of localized region, the
semi-classical method developed by Zhu and Nakamura3,4 could be used to deal
non-adiabatic coupling by equation (2.6) and (2.7). Now The question is how to make
sure the quality of adiabaticity, that is, how to choose a coordinate that improve the
adiabaticity. Previously, Jacobi5 (non-adiabatic coupling did not die out in asymptotic
region), hyper-spherical 6 and hyper-spherical elliptic coordinates7 are being used. But
there is no analytical form for more than 5 atoms, the work become tedious even for
obtaining the adiabatic curves only. A general coordinate is needed for this
semi-classical method, since the main issue is to obtain the adiabatic curves, and the
following steps will be just like a service pattern. So how to choose this coordinate, in
fact, it’s min energy path (MEP), which has been developed thoroughly. How can min
energy path be an appropriate slow variable will be explained in 2-2.1. Then, I would
like to introduce how the semi-classical theory developed by Zhu and Nakamura3,4
can be applied to this coordinate in section 2-2.2.
2-2.1 Min energy path
In order to explain why MEP could be proper slow variable for adiabatic
approximation, knowing the basics of MEP is necessary. Let’s recall the fundamental
concept for min energy path, there would be 3N-7 mode with gradient equal to zero,
orthogonal to the min energy path which may contains non zero gradient. So for the
ζ ζ ζ + ⋅ ⋅ ⋅ ∂ ∂ + = k a a v a v x v 2 1 ) ( ) ( ) ( (2.68)
where x is arbitrary point; x = {xir}, a is a point on the reaction path; a = {air}, k =
{kir,i’r’} with ; ; ; ' 2 , r i ir r i ir x x v k ∂ ∂ ∂ = (2.69)
, and ζ is the displacement vector. Here i = 1, 2, 3…..,N , r = x, y, z. For
displacementζ that is orthogonal in the 3N dimensional vector space component of
the gradient part in the equation (2.68). In order to get the normal modes for vibration
that are orthogonal to the reaction path, it is also necessary for ζ to orthogonal to
the 3N space vector component of the rotation and translation for the complete
N-atom system. Since it’s not the case of minimum or saddle point, it is necessary to
project out the infinitesimal rotation, translation and the unit vector along the reaction
path, otherwise there will contamination of rotation and translation during normal
analysis. So at each point, one define projected force constant matrix kP for normal
analysis as ) 1 ( ) 1 ( P k P kP = − ⋅ ⋅ − (2.70) After doing normal analysis at harmonic approximation, one could expect there will
be 3N-7 nonzero eigenvalues; which give frequencies orthogonal to the reaction path,
seven zero with six correspond to rotation and translation, one for reaction coordinate.
∑
= − = − = + 7 3 1 2 2 0 7 3 3 2 1 ( ) 2 1 ) ( ) ..., , , , ( k N k k k N v s w s Q Q Q Q Q s v (2.71)where s is reaction path ,Qk is the normal coordinate orthogonal to s and v0 is the
potential energy on the reaction path. The classical one-dimension motion
Hamiltonian along reaction path for 3N-7>2 has been reported as
∑
= − = + + + = 3 7 1 0 2 ) 2 1 )( ( 2 1 ) ( ) ( 2 1 ) ( k N k k k s s n p s p A s v s w s n H (2.72)by Miller8, where A(s) is the correction factor and ps is the momentum along reaction
coordinate. From equation (2.72), we can easily find that the first term is equal to the
kinetic energy of slow variable in equation (2.2), the second and third term then is
correspond to adiabatic potential, which is the En in (2.5).This term (En) is only
information needed for semi-classical theory. Up to now, I have not explained why
reaction path could be the right coordinate to choose. Since the reaction coordinate
separate reactant and product into two regions, there’s no problem of non-adiabatic
coupling between reactants and products in asymptotic region as Jocabi coordinate
does. And the reactive transitions mostly occur around the saddle point (transition
state). Besides that, the way of obtaining the adiabatic curves (En) is much more
general than Jocabi, hyper-spherical and hyper-spherical elliptic coordinate, since the
theory for normal mode analysis on reaction coordinate8 has been developed
completely. En here is correspond to J = 0 adiabatic curves, since equation (2.72) only
during the reaction, so the information of chemical dynamics preserved in constant J,
that is the dynamics won’t be mixed between each J. But in order to obtain the total
dynamics, the message of J>0 is needed. The J-shift approximation11,12,13, which
relates the state to state reaction probability for arbitrary J and its body-fixed
projection k to the one actually calculated for J = 0:
) ( ) ( 0 *, , K J J f i K J f i E P E E P→ ≈ →= − (2.73) , will be used to gain case for J > 0, where E*J,K is the rotational energy of transition
state.
2-2.2 Semi-classical theory
As mentioned above, MEP divides the reactant and product into two different
regions; no problem of non-adiabatic coupling in the asymptotic region, and reactive
transitions mainly occur around saddle point (transition), so it make it possible to treat
the chemical reaction in ordinary scattering problem. That is to say, we can apply the
semi-classical theory of non-adiabatic transition developed by Zhu and Nakamura9,10.
2-2.2.1 Zhu and Nakamura theory
The theoretical studies of non-adiabatic of transitions between potential energy
published the pioneering paper independently. They have shown the non-adiabatic
transition probability at curve crossing point between two curves :
) 2 exp( 2 F v E PLZ h ∆ − = π (2.74)
, so called famous Landau-Zener formula, where ∆E is the diabatic coupling:
2 1 2 W W E = − ∆ (2.75) and F is the difference of the slopes of the two diabatic potential (V1 and V2) at Rx,
which is the crossing point between V1 and V2 as shown in figure 2.1. But there are
five general defects for this formula, which are summarized17 as follows: (1) Not
work at energies near and lower than the crossing point. (2) No good formula exists
for transmission when the two diabatic curves cross with different signs of slopes. (3)
The available accurate formulas, which are valid only at energies higher than the
crossing point, contain inconvenient complex contour integrals and are not very useful
for experimentalists. (4) The Landau-Zener formula requires the knowledge of
diabatic potential, which can’t be uniquely obtained from adiabatic potentials. (5) The
accurate phases to define the scattering matrixs are not available for all cases. In this
report, we would take advantages of point (2), (3) and (4), that is, we only consider
the transition point lower than total energy and no phase consideration. The main
purpose of this report is to check whether taking MEP as the slow variable is valid or
2-3 Application to adiabatic curves on MEP
Since the properties of MEP, the adiabaticity holds in localized region. Then the
most important non-adiabatic transition occur between two adjacent adiabatic curves.
(Non-adiabatic transition between non-adjacent may contribute a little, but not that
important, which may be considered after obtaining the diabatic curves from adiabatic
curves as ref 1 does. This won’t be shown in this report.) So let us look at the easiest
case; two adiabatic curves, as Figure 2.2. The scattering wave function for asymptotic
region can be written in WKB form:
) (s ) ) ( exp( ) ( ) ) ( exp( ) ( ) ( = + − →∞ Ψ
∫
∫
s T n n n s T n n n R R n R n ds s k i s k B ds s k i s k A s (2.76) ) (s ) ) ( exp( ) ( ) ) ( exp( ) ( ) ( = + − → −∞ Ψ∫
∫
s T n n n s T n n n L L n L n ds s k i s k D ds s k i s k C s (2.77),where TLn and TRn are the same for transition point lower than the total energy
considered, but left-side turning point and right-side turning point for tunneling case,
and )) ( ( 2 ) (s E En s kn =
µ
− . (2.78) The scattering matrix is defined by = = = 2 1 2 1 41 43 42 41 34 33 32 31 24 23 22 21 14 13 12 11 2 1 2 1 44 34 24 14 43 33 23 13 42 32 22 12 41 31 21 11 2 1 2 1 2 1 2 1 C C B B S S S S S S S S S S S S S S S S C C B B S S S S S S S S S S S S S S S S C C B B s D D A A (2.79)
, where the outgoing coefficients (purple one in the figure 2.2) are represented by the
incoming coefficients (black one in the figure 2.2), and Sij=Sji. It’s not unexpected to
obtain S matrix in the form of equation (2.79). Since the outgoing part is contributed
by the outgoing part, that is to say, Sij itself is related to the probability that outgoing
coefficient formed from the incoming coefficient. In fact, it is in the form of square
root of probability with phase; because it is the square of coefficient that gives the
meaning of probability not coefficient itself. Take A1 for example:
2 41 1 31 2 21 1 11 1 S B S B S C S C A = + + + (2.80) , where S11 is related to the probability of the reflection of the incoming wave B1, S12
is the probability of the reflection of the incoming wave of B2 which transmit to lower
curve, S31 is the probability that C1 stay at lower curve and S41 is the probability that
C2 stay at upper curve. In this model, there two kinds of S matrix:
1. Non-adiabatic transition between two curves, so called I matrix ; like point I1
and point I2 in figure 2.3:
0 0 1 0 0 1 1 0 0 1 0 0 2 1 2 1 1 1 1 1 1 1 1 1 2 1 2 1 1 1 1 1 − − − − − − == − − C C B B e p p p e p e p p p e p D D A A ix ix ix ix
Or 1 1 T with 0 0 1 1 1 1 1 1 1 2 1 2 1 1 1 2 1 2 1 1 2 1 2 1 − − − = = = −ix ix e p p p e p C C B B T T C C B B I D D A A 1 I for (2.81) 0 0 1 0 0 1 1 0 0 1 0 0 ' 2 ' 1 ' 2 ' 1 2 2 2 2 2 2 2 2 ' 2 ' 1 ' 2 ' 1 2 2 2 2 − − − − − − == − − C C B B e p p p e p e p p p e p D D A A ix ix ix ix Or 1 1 0 0 2 2 2 2 2 2 2 ' 2 ' 1 ' 2 ' 1 2 2 ' 2 ' 1 ' 2 ' 1 2 ' 2 ' 1 ' 2 ' 1 − − − = = = −ix ix e p p p e p T with C C B B T T C C B B I D D A A 2 I for (2.82) , where p1 and p2 are the probability of the transition between lower curve and
upper curve at position of I1 and I2
2. Tunneling; like the E1 case in figure 2.3:
− − − − − − = − − 2 1 ' 2 ' 1 2 2 1 1 2 2 1 1 ' 2 ' 1 2 1 2 2 2 1 1 1 2 2 2 1 1 1 0 1 0 0 0 1 1 0 0 0 1 0 D D A A p e p e e p e p e e p e e p e p e e p e B B C C t i t ix i t i t ix i t ix i t i t ix i t i t t t t t t t t t t t t φ φ φ φ φ φ φ φ (2.83)
, where pt1 and pt2 represents the probability of transmittance at lower and upper curve
for energy equals to E1. These two kinds shown above are only single S matrix. In
order to obtain the final s matrix; which connects incoming and outgoing of the very
beginning and end, one should know how to link the multiple s matrix. From figure
