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
Ψ the wave function of nucleuses. Since the state-to-state dynamics here are Nux
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 methods; quasi-classical trajectory (QCT), quantum and semi-classical methods.
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
with total wave function to be:
) obtain the non-adiabatic coupling terms as equation (2.6) and equation (2.7):
) pretty small, that is, we could omit these effects and obtain
0 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 system? From the view of physical meanings, since the motion of nuclei is much
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
, where A, B refer to nuclie and i, j refer to electrons. The Hamiltonian may be written explicitly as 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
)
Or And the total schrondinger equation of the molecular system would be
0 , 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
∑
Then we obtain
[
2]
0A more compact notation is needed for equation (2.20), so we introduced the following quantities used by Tully2
r
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,∇Aϕneis much smaller than∇AFnNuc. So we could obtain
And total wave function would be ) 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;
Hartree–Fock or self-consistent field (SCF) method and Post–Hartree–Fock, where
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
)
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
)
B WhereHˆ is the form as equation (2.13) or (2.14) which could be rewritten as e
∑ ∑
If Ne equals to 2N, and the form of Slater determinant changed to
)
Then equation (2.29) can be further written as
∑∑
Where the third term of equation (2.30) is not considered, since it is just an effect of a
constant and shifts to the eigenvalues and
i
m
JB is called coulomb integral and the
j iB
KB 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
F is the Fock operator, which is given by
[ ]
J k is the coulomb operator given by
K m
K k is the exchange operator given by
i m
So the expression for the ith molecular orbital would be
i j
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 between
electrons 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 ϕMOe as the
linear combinations of basis functionsφv, 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 rj
withφµ*(rj). Then we can get
k c
S c
F
k
v v v k
v v
v =
∑
=1 ,2 ,L,∑
µ ε µ µ (2.46) WhereFµvis given byrj
j v j j
v r F r r
Fµ = φµ( ) )( )φ ( ) (2.47) andSµvis given by
rj
j v j
v r r
Sµ = φµ( )φ ( ) (2.48) Ifφvis chosen a set of real functions; which is general the case. Then bothFµvandSµv are 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
several advantages above, but there exist one big defect, that is, the independent
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
ri
F
Hˆ(0) ( ) (2.52)
In whichF(ri)is the Fock operator
∑
=
i a
i i
E(0) ε (2.53) andΦ(i0) 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
∑∑
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 So considering to second order energy (MP2), the energy would be
∑ ∑∑ ∑∑
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
the following form
L
excitation as bellow, where the coefficients are not obtained by variation method
∑∑
ΦSo equation (2.59) would be
(
1+ ˆ1+ ˆ2 + ˆ3 +)
Φ0, which is the exact solution to the equation (2.17). By multiplying ( )T
e− ˆ and integrate withΦα, which is at least single excitation, then we can get the information of coefficient in it as bellow
ˆ 0
And CCSD and CCSD(T) is the truncated coupled-cluster method, where CCSD take
( )T
e ˆ as bellow ( )
2 1
ˆ 1 Tˆ Tˆ
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 adiabatic
electronic 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
with this case. The information needed is the En in equation (2.5) without solving the
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
general point one could has
ζ ζ
ζ + ⋅ ⋅
∂ ⋅ +∂
= k
a a a v
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 ir i r
i
ir x x
k v
∂
∂
= ∂ (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.
So this provides the following approximated potential surface,
∑
== −potential energy on the reaction path. The classical one-dimension motion
Hamiltonian along reaction path for 3N-7>2 has been reported as
∑
== − +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
contains vibrational part, where J is the total angular momentum, which is conserved
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
that is the dynamics won’t be mixed between each J. But in order to obtain the total