- Email: [email protected]
The diabatic decoupling approximation
Volume 27, number 2
CHEMICAL PHYSICS LETTERS
THE DIABATIC
DECOUPLING
15 July 1974
APPROXIMATION
B.R. JOHNSON Department
of Chemistry, The Ohio State University, Columbus, Ohio 43210, USA
Received 8 April 1974
A new representation having desirable features of both the adiabatic and diabatic representations is defied. The states of a multichannel problem are partitioned. The states within a partition are diabatically coupled by an effective symmetric potential matrix, but are decoupled between partitions in a manner similar to the Born-Oppenheimer approximation. A sample problem involving curve crossing is presented.
In this paper we are concerned with problems of either a scattering or bound state nature which can, to some approximation, be reduced to the solution of the following N state matrix Schrodinger equation [d2/dx2 + V(x)]\k(x)
= 0,
(1)
where V(x) = (2/J/F?)[El
- V(x) - E] .
(2)
E is the total energy, ~1is the reduced mass, V(x) is a
symmetric potential matrix which approaches zero as x approaches infinity, E is a diagonal matrix whose elements are the threshold energies of the various channels, and I is the unit matrix. This equation can be transformed to other equivalent representations by an orthogonal transformation. Let C(x) be a general x dependent, orthogonal matrix. Multiply eq. (1) by CT(x) (superscript T means transpose) and make use of the orthogonality relation
CT(x)C(x)
=I
(3)
to obtain [d2/dx2 + U(x)]@(x)
= ]A(x)dlb
+ B(x)]@(x)
, (4)
where we have defined %x) = CT(x)W)
3
U(x) = CT(x) V(x) C(x) ,
(5)
(6)
A(x) = -2CT(x)dC(x)/ti
(7)
and B(x) = -CT(x)d2C(x)/dx2
.
(8)
No approximations have been made so far, and eq. (4) is equivalent to eq. (1). Two representations are of particular importance in the literature: the diabatic and the adiabatic [ 11. The diabatic representation is one in which the matrices A and B are both zero. It is obvious from this that eq. (1) is written in a diabatic representation. This is usually the most convenient for numerical coupled channel calculations. The adiabatic representation is defined to be the one in which the matrix U is diagonal. The elements of U are the eigenvalues of the symmetric matrix V and can be obtained along with the orthogonal matrix C by solving the eigenvalue problem represented by eq. (6). Since U is diagonal, the coupling between channels in eq. (4) is provided by A and B. The matrix U is proportional to the mass ,u [see eqs. (2) and (6)] while the matrices A and B are not. Therefore in the large mass limit (for non-curve crossing problems) U will be much larger than A and B so that to a good approximation A and B can be neglected thus decoupling the problem. This is the basis of the BornOppenheimer approximation [2] which works very well for many problems in molecular physics. 289
Volume
27, number
2
CHEMICAL
PHYSICS
In this paper we will be concerned with problems for which adiabatic decoupling is not a good approximation; in particular we have in mind the curve crossing problem in which the M diagonal elements Vii(x), i G M, are interconnected by crossing points, but do not cross any of the diagonal elements Vii(x) for M < i < N. Transforming to the adiabatic representation, it is found that the matrix elements of A and B which couple the states @‘i,i GM, are large and non-negligible in small regions around the crossing points. Thus these states remain strongly coupled even in the adiabatic representation. One advantage we have gained however is that the states aj, i d M, are approximately decoupled from the remaining states @‘i,i > M. The problem has thus been effectively reduced to an M state problem in the adiabatic representation which we could then solve. Although this is possible [3] it turns out in practice to be difficult for several reasons. First, the matrices A and B must be calculated, which means that the first and second derivatives of the matrix C(x) must be calculated [see eqs. (7) and (S)]. Second, the matrix elements of A and B are usually highly peaked functions around the crossing points, and so care must be exercised when integrating across these narrow peaks. Finally, a first derivative term appears in the adiabatic equations [see eq. (4)], which means that less efficient numerical integration techniques must be used than are available for the matrix Schrijdinger equation in the diabatic representation. A better method is the diabatic decoupling representation which shares with the adiabatic representation the desirable feature of approximately decoupling the states Qi, id M, from the states i > M. It has the additional feature however of casting the remaining M state problem in a diabatic form with an effective symmetric diabatic potential matrix, thus making the solution much easier. The diabatic decoupling representation is most easily defined by first transforming to the adiabatic representation and then transforming from this to the diabatic decoupling representation. The idea behind this method is to decouple the problem as much as possible in the adiabatic representation. This allows us approximately to decouple the group of states pi, i GM, from the group of states @‘i,M < i < N. We then try to transform as much as possible back to the diabatic representation within each group of states. 290
LETTERS
15 July 1974
Eq. (6), written in partitioned
(!ll
form, is
;,,>
Quantities with a tilde over them_are iE the adiabatic representation. The submatrices U 11, C 11 and V1 I have dimensions M X M. Although the complete matrix e is orthogonal, the submatrices E1 I and &2 do not have any particular properties except that they are square. Any square matrix can be written in polar form, i.e., as the product of a symmetric matrix times an orthogonal matrix [4]. Thus, we can write =S,Q,
Ll
(n = 1,2) ,
(10)
where S, is a symmetric matrix and 0, is an orthogonal matrix. The matrices S, and Q, are easily evaluated. Multiply eq. (10) from the right by its transpose and take the positive square root to obtain s, = [Enn E;m]r’a
(n= 1,2).
(11)
By positive square root, we mean that the eigenvalues of S, are positive. The matrix Q, can then be computed using eq. (10). The orthogonal matrix Q is defined to be
E2).
Q=@
(12)
By definition, the transpose of this matrix, QT, transforms the adiabatic representation to the diabatic decoupling representation. Thus, the matrix C which transforms from the diabatic to the diabatic decoupling representation is C=EQT
(13)
and the potential given by
matrix U, defined by eq. (6) is
U=QcQT.
.
(14)
By partitioning the matrices in eq. (14), the submatrices U,, of U are Unn =
QAlnQ~
(n= 1,2).
The coupled channel Schriidinger
(15) equation
in the
CHEMICAL PHYSICS LETTERS
Volume 21, number 2
Table 1 Comparison of exact, two-state and diabatic decoupling approximation values of the S-matrix for several values of I 1
Is111 ISlzl
ArgGll)
Arg(Sd
Arg(S22)
(exact three state) 0 90 180 270 360
0.885 0.884 0.983 0.944 1.000
0.465 0.468 0.184 0.329 0.008
0.690 0.316 0.446 1.575 2.137
3.944 2.542 5.822 5.034 5.766
4.056 1.626 1.773 5.351 6.253
(two state) 0 0.982 90 0.993 180 0.936 270 0.982 360 1.000
0.186 0.122 0.352 0.188 0.012
0.865 1.421 3.608 6.220 1.656
2.846 2.004 0.226 3.757 5.251
1.686 5.729 6.268 4.436 5.705
0.686 0.312 0.446 1.576 2.137
3.942 2.540 5.821 5.034 5.766
4.057 1.627 1.771 5.350 6.253
(diabatic decoupling) 0 90 180 270 360
0.885 0.884 0.984 0.946 1.000
0.465 0.467 0.178 0.325 0.008
diabatic decoupling representation is given by eq. (4) where the matrices C(x) and U(x) are defined by eqs. (13) and (14). In the large mass limit, A and B are small compared to U, and if A and B are set equal to zero, the states aj, i GM, are diabatically coupled by the symmetric potential matrix U 11(x) but are decoupled from the remaining states Qj, i > M. Although we have not been able to prove that our definition of the matrix 0 by eqs. (IO)-( 12) is the best, we believe it is. By “best”, we mean that this choice of Q produces a matrix C, defined by eq. (13) which in some overall sense has the smallest derivatives and thus produces the smallest A and B matrices. In numerical tests conducted so far, any other choice for Q has produced v~orse results. As an example of the use of this method consider the following three-channel scattering problem. The diabatic potential matrix in atomic units is V,, + 15, =(21.1/x)exp(-x/0.678)+Z(Z+1)/2~2, V22 + E, =(21.1/x-12.1) + Z(Z+1)/2~2
exp(-x/0.678)
+ 0.6174 ,
v33 t E, = V,, + El to.753
V,, = 0.5 exp (-x/0.667)
15 July 1974
, V,, = 4V,,
, V23 = 3V,,
The mass is cc = 6089 au and the energy is E = 2.6 au. The diagonal potential curves 1 and 2 cross each other but neither of them cross curve 3. This 3-state potential has no particular physical significance and is just an ad-hoc invention for the sample problem. The potentials for channels 1 and 2 are the same as Olson and Smith [.5] used for He+-Ne scattering although the strength of VI2 has been increased. The reduced mass is also the same as for the He-Ne system. An exact three-state diabatic calculation was done for several 1 values and the results are given in table 1. Only the S-matrix elements, S, 1, S,, , and S,,, are printed in the table since they are the only ones needed for comparison with the decoupling approximations. The magnitudes IS,, 1and ]S23] are of the order 10e5 and IS,,1 x 1 .O. To the accuracy of the table IS,,1 = IS,,], therefore IS,,] is not printed. The log derivative method was used to do this calculation [6]. Two decoupling approximations are tested. The first and most elementary method of decoupling is to set VI3 and V23 to zero, this is just equivalent to retaining the first two diabatic states and will be called the two-state approximation. The second is the diabatic decoupling approximation in which the matrices A and B in the diabatic decoupling representation are set equal to zero. The results are given in table 1. Comparing these to the exact results, we see that the two-state approximation is poor whereas the diabatic decoupling approximation is quite good. Some applications and a generalization of the diabatic decoupling approximation will appear in future publications.
References [l] F.T. Smith, Phys. Rev. 179 (1969) 111. [2] J.O. Hirschfelder and W.J. Meath, in: Advances in chemical physics, Vol. 12, ed. J.O. Hirschfelder (Wiley, New York, 1967). [3] S.A. Evans, J.S. Cohen and N.F. Lane, Phys. Rev. A4 (1971) 2235,2248. [4] M.C. Pease III, Methods of matrix algebra (Academic Press, New York, 1965) p. 102. [5] R.E. Olson and F.T. Smith, Phys. Rev. A3 (1971) 1607 (Errata, Phys. Rev. A6 (1972) 526). [6] B.R. Johnson, J. Comp. Phys. 13 (1973) 445.
291
CHEMICAL PHYSICS LETTERS
THE DIABATIC
DECOUPLING
15 July 1974
APPROXIMATION
B.R. JOHNSON Department
of Chemistry, The Ohio State University, Columbus, Ohio 43210, USA
Received 8 April 1974
A new representation having desirable features of both the adiabatic and diabatic representations is defied. The states of a multichannel problem are partitioned. The states within a partition are diabatically coupled by an effective symmetric potential matrix, but are decoupled between partitions in a manner similar to the Born-Oppenheimer approximation. A sample problem involving curve crossing is presented.
In this paper we are concerned with problems of either a scattering or bound state nature which can, to some approximation, be reduced to the solution of the following N state matrix Schrodinger equation [d2/dx2 + V(x)]\k(x)
= 0,
(1)
where V(x) = (2/J/F?)[El
- V(x) - E] .
(2)
E is the total energy, ~1is the reduced mass, V(x) is a
symmetric potential matrix which approaches zero as x approaches infinity, E is a diagonal matrix whose elements are the threshold energies of the various channels, and I is the unit matrix. This equation can be transformed to other equivalent representations by an orthogonal transformation. Let C(x) be a general x dependent, orthogonal matrix. Multiply eq. (1) by CT(x) (superscript T means transpose) and make use of the orthogonality relation
CT(x)C(x)
=I
(3)
to obtain [d2/dx2 + U(x)]@(x)
= ]A(x)dlb
+ B(x)]@(x)
, (4)
where we have defined %x) = CT(x)W)
3
U(x) = CT(x) V(x) C(x) ,
(5)
(6)
A(x) = -2CT(x)dC(x)/ti
(7)
and B(x) = -CT(x)d2C(x)/dx2
.
(8)
No approximations have been made so far, and eq. (4) is equivalent to eq. (1). Two representations are of particular importance in the literature: the diabatic and the adiabatic [ 11. The diabatic representation is one in which the matrices A and B are both zero. It is obvious from this that eq. (1) is written in a diabatic representation. This is usually the most convenient for numerical coupled channel calculations. The adiabatic representation is defined to be the one in which the matrix U is diagonal. The elements of U are the eigenvalues of the symmetric matrix V and can be obtained along with the orthogonal matrix C by solving the eigenvalue problem represented by eq. (6). Since U is diagonal, the coupling between channels in eq. (4) is provided by A and B. The matrix U is proportional to the mass ,u [see eqs. (2) and (6)] while the matrices A and B are not. Therefore in the large mass limit (for non-curve crossing problems) U will be much larger than A and B so that to a good approximation A and B can be neglected thus decoupling the problem. This is the basis of the BornOppenheimer approximation [2] which works very well for many problems in molecular physics. 289
Volume
27, number
2
CHEMICAL
PHYSICS
In this paper we will be concerned with problems for which adiabatic decoupling is not a good approximation; in particular we have in mind the curve crossing problem in which the M diagonal elements Vii(x), i G M, are interconnected by crossing points, but do not cross any of the diagonal elements Vii(x) for M < i < N. Transforming to the adiabatic representation, it is found that the matrix elements of A and B which couple the states @‘i,i GM, are large and non-negligible in small regions around the crossing points. Thus these states remain strongly coupled even in the adiabatic representation. One advantage we have gained however is that the states aj, i d M, are approximately decoupled from the remaining states @‘i,i > M. The problem has thus been effectively reduced to an M state problem in the adiabatic representation which we could then solve. Although this is possible [3] it turns out in practice to be difficult for several reasons. First, the matrices A and B must be calculated, which means that the first and second derivatives of the matrix C(x) must be calculated [see eqs. (7) and (S)]. Second, the matrix elements of A and B are usually highly peaked functions around the crossing points, and so care must be exercised when integrating across these narrow peaks. Finally, a first derivative term appears in the adiabatic equations [see eq. (4)], which means that less efficient numerical integration techniques must be used than are available for the matrix Schrijdinger equation in the diabatic representation. A better method is the diabatic decoupling representation which shares with the adiabatic representation the desirable feature of approximately decoupling the states Qi, id M, from the states i > M. It has the additional feature however of casting the remaining M state problem in a diabatic form with an effective symmetric diabatic potential matrix, thus making the solution much easier. The diabatic decoupling representation is most easily defined by first transforming to the adiabatic representation and then transforming from this to the diabatic decoupling representation. The idea behind this method is to decouple the problem as much as possible in the adiabatic representation. This allows us approximately to decouple the group of states pi, i GM, from the group of states @‘i,M < i < N. We then try to transform as much as possible back to the diabatic representation within each group of states. 290
LETTERS
15 July 1974
Eq. (6), written in partitioned
(!ll
form, is
;,,>
Quantities with a tilde over them_are iE the adiabatic representation. The submatrices U 11, C 11 and V1 I have dimensions M X M. Although the complete matrix e is orthogonal, the submatrices E1 I and &2 do not have any particular properties except that they are square. Any square matrix can be written in polar form, i.e., as the product of a symmetric matrix times an orthogonal matrix [4]. Thus, we can write =S,Q,
Ll
(n = 1,2) ,
(10)
where S, is a symmetric matrix and 0, is an orthogonal matrix. The matrices S, and Q, are easily evaluated. Multiply eq. (10) from the right by its transpose and take the positive square root to obtain s, = [Enn E;m]r’a
(n= 1,2).
(11)
By positive square root, we mean that the eigenvalues of S, are positive. The matrix Q, can then be computed using eq. (10). The orthogonal matrix Q is defined to be
E2).
Q=@
(12)
By definition, the transpose of this matrix, QT, transforms the adiabatic representation to the diabatic decoupling representation. Thus, the matrix C which transforms from the diabatic to the diabatic decoupling representation is C=EQT
(13)
and the potential given by
matrix U, defined by eq. (6) is
U=QcQT.
.
(14)
By partitioning the matrices in eq. (14), the submatrices U,, of U are Unn =
QAlnQ~
(n= 1,2).
The coupled channel Schriidinger
(15) equation
in the
CHEMICAL PHYSICS LETTERS
Volume 21, number 2
Table 1 Comparison of exact, two-state and diabatic decoupling approximation values of the S-matrix for several values of I 1
Is111 ISlzl
ArgGll)
Arg(Sd
Arg(S22)
(exact three state) 0 90 180 270 360
0.885 0.884 0.983 0.944 1.000
0.465 0.468 0.184 0.329 0.008
0.690 0.316 0.446 1.575 2.137
3.944 2.542 5.822 5.034 5.766
4.056 1.626 1.773 5.351 6.253
(two state) 0 0.982 90 0.993 180 0.936 270 0.982 360 1.000
0.186 0.122 0.352 0.188 0.012
0.865 1.421 3.608 6.220 1.656
2.846 2.004 0.226 3.757 5.251
1.686 5.729 6.268 4.436 5.705
0.686 0.312 0.446 1.576 2.137
3.942 2.540 5.821 5.034 5.766
4.057 1.627 1.771 5.350 6.253
(diabatic decoupling) 0 90 180 270 360
0.885 0.884 0.984 0.946 1.000
0.465 0.467 0.178 0.325 0.008
diabatic decoupling representation is given by eq. (4) where the matrices C(x) and U(x) are defined by eqs. (13) and (14). In the large mass limit, A and B are small compared to U, and if A and B are set equal to zero, the states aj, i GM, are diabatically coupled by the symmetric potential matrix U 11(x) but are decoupled from the remaining states Qj, i > M. Although we have not been able to prove that our definition of the matrix 0 by eqs. (IO)-( 12) is the best, we believe it is. By “best”, we mean that this choice of Q produces a matrix C, defined by eq. (13) which in some overall sense has the smallest derivatives and thus produces the smallest A and B matrices. In numerical tests conducted so far, any other choice for Q has produced v~orse results. As an example of the use of this method consider the following three-channel scattering problem. The diabatic potential matrix in atomic units is V,, + 15, =(21.1/x)exp(-x/0.678)+Z(Z+1)/2~2, V22 + E, =(21.1/x-12.1) + Z(Z+1)/2~2
exp(-x/0.678)
+ 0.6174 ,
v33 t E, = V,, + El to.753
V,, = 0.5 exp (-x/0.667)
15 July 1974
, V,, = 4V,,
, V23 = 3V,,
The mass is cc = 6089 au and the energy is E = 2.6 au. The diagonal potential curves 1 and 2 cross each other but neither of them cross curve 3. This 3-state potential has no particular physical significance and is just an ad-hoc invention for the sample problem. The potentials for channels 1 and 2 are the same as Olson and Smith [.5] used for He+-Ne scattering although the strength of VI2 has been increased. The reduced mass is also the same as for the He-Ne system. An exact three-state diabatic calculation was done for several 1 values and the results are given in table 1. Only the S-matrix elements, S, 1, S,, , and S,,, are printed in the table since they are the only ones needed for comparison with the decoupling approximations. The magnitudes IS,, 1and ]S23] are of the order 10e5 and IS,,1 x 1 .O. To the accuracy of the table IS,,1 = IS,,], therefore IS,,] is not printed. The log derivative method was used to do this calculation [6]. Two decoupling approximations are tested. The first and most elementary method of decoupling is to set VI3 and V23 to zero, this is just equivalent to retaining the first two diabatic states and will be called the two-state approximation. The second is the diabatic decoupling approximation in which the matrices A and B in the diabatic decoupling representation are set equal to zero. The results are given in table 1. Comparing these to the exact results, we see that the two-state approximation is poor whereas the diabatic decoupling approximation is quite good. Some applications and a generalization of the diabatic decoupling approximation will appear in future publications.
References [l] F.T. Smith, Phys. Rev. 179 (1969) 111. [2] J.O. Hirschfelder and W.J. Meath, in: Advances in chemical physics, Vol. 12, ed. J.O. Hirschfelder (Wiley, New York, 1967). [3] S.A. Evans, J.S. Cohen and N.F. Lane, Phys. Rev. A4 (1971) 2235,2248. [4] M.C. Pease III, Methods of matrix algebra (Academic Press, New York, 1965) p. 102. [5] R.E. Olson and F.T. Smith, Phys. Rev. A3 (1971) 1607 (Errata, Phys. Rev. A6 (1972) 526). [6] B.R. Johnson, J. Comp. Phys. 13 (1973) 445.
291