Baer M., Billing G.D. (eds.) - The role of degenerate states in chemistry (Adv.Chem.Phys. special issue, Wiley, 2002)

312

aron kuppermann and ravinder abrol

^

2

ð lÞ is the grand canonical angular momentum operator

and

^2

^2

^Il2

4Jz

ð lÞ ¼ oðy; flÞ þ

cos2y

2

^2

^Il2

^Il2

^Il2

J^Il2

J

Jz

Jþ

þ J

þ

"

þ

z

#

1 þ siny

2

4

1

^2

^Il2

^Il2

^Il2

J

Jz

Jþ

þ J

þ

"

#

sin2 y

2

4

cosy

Il

Il

q

^

^

2h sin2 y ðJþ

J

Þ qfl

where

^2

2

1

q

q

1

q2

!

o

ðy; flÞ ¼ 4h

sin2y

þ

sin2y

qy

qy

sin2y

qfl2

and

^Il

^Il

^Il

J

¼ Jx

iJy

ð84Þ

ð85Þ

ð86Þ

^Il	^Il		^Il		^
Jx	, Jy	, and Jz		are the components of the total orbital angular momentum J of
				^2	^Il	,
the nuclei in the Il frame. The Euler angles al; bl; cl appear only in the J					, Jz
	^Il		angular momentum operators. Since the results of their operation on
and J

Wigner rotation functions are known, we do not need their explicit expressions in terms of the partial derivatives of those Euler angles.

B.Partial Wave Expansion

In the two-adiabatic-electronic-state Born–Huang description of the total orbital wave function, we wish to solve the corresponding nuclear motion Schro¨dinger equation in the diabatic representation

	h2	ð87Þ
	2m fIrR2 l þ Wð2ÞdðqlÞg þ fedðqlÞ EIg vdðRlÞ ¼ 0

for the diabatic orbital nuclear wave function column vector vdðRlÞ

d	R	w1d	ðRl	Þ	ð	88	Þ
v ð	lÞ ¼	w2d	Rl	!
			ð	Þ

In Eq. (87), the gradient term containing the transverse coupling has been dropped because its inclusion in this formalism leads to numerical inefficiencies

quantum reaction dynamics for multiple electronic states 313

in the very efficient logarithmic derivative propagator [87,88] used in solving Eq. (103). In the process of obtaining the ADT matrix, the magnitude of the transverse coupling vector is minimized over the entire internal nuclear configuration space following the procedure described in Section III. This makes dropping the gradient term a very good approximation. After the diabatization, since we know the transverse coupling vector, the effect of the gradient term on the scattering results obtained without it can be assessed using perturbative or other methods. In Eq. (87), edðqlÞ is a 2 2 diabatic energy matrix

e

d

ð

q

lÞ ¼

e11d

ð

q

lÞ

e12d

ð

q

lÞ

ð

89

Þ

d

d

e12

ðqlÞ

e22

ðqlÞ

and Wð2ÞdðqlÞ is a 2 2 second-derivative diabatic coupling matrix

Wð2Þd

ð

q

lÞ ¼

w1d;1ðqlÞ

w1d;2ðqlÞ

ð

90

Þ

w2d;1ðqlÞ

w2d;2ðqlÞ

i; j

ð

¼

1; 2

Þ

2

l

only rather than on the full

The Wð2Þd and wð2Þd

i; j

now depend on q

Rl. The reason is as follows. The rRl appearing in the three body and two-

electronic-state version of Eq. (36) contains terms in r; y; fl as well as in the
a	Il	^Il2	^Il2 ^Il2	, which
		, the latter through the angular momentum operators Jx	, Jy , Jz
are the squares of the components of the total angular momentum vector				^
				J in the

principal axes of inertia frame that also appeared in Eq. (61). Since, as discussed in Section II.A, celi ;ad and therefore celi ;d ði ¼ 1; 2Þ depend only on ql (rather on the full Rl), the result of the application of those angular momentum operators on these diabatic electronic wave functions is zero. Therefore, the only contributions to r2Rl celi ;dðr; qlÞ come from the terms in r2Rl that contain ql

only, which is different from the			first-derivative	$Rl	coupling	elements for
~	Il	Þ factor in the	right-hand side	of	Eq. (60)	results in a
which the Rða

dependence of Wð1Þd on aIl when using Eq. (61).

Since the second-derivative coupling matrix Wð2Þd is only an additive term in Eq. (87), we can merge it with the diabatic energy matrix and define a 2 2

diabatic matrix

d

q

d

q

h2

Wð2Þd

q

91

ð

lÞ ¼ e

ð

ð

ð

Þ

e

lÞ 2m

lÞ

By using Eq. (91), we can rewrite Eq. (87) as

h2

þ fedðqlÞ EIg vdðRlÞ ¼ 0

IrR2 l

ð92Þ

2m

314	aron kuppermann and ravinder abrol

The two diabatic nuclear wave functions wd1 and wd2 can be expressed as linear combinations of auxiliary nuclear wave functions wd1;JM and wd2;JM , respectively (the linear combinations referred to as partial wave expansions and the individual wd1;JM and wd2;JM referred to as partial waves), such that if we define another nuclear wave function column vector

			vd;JM ðRlÞ ¼	w1d;JM Rl			Þ !	ð93Þ
				wd;JM	ðR	l
				2	ð		Þ
then v	d;JM						^ nu	(given
			is a simultaneous eigenfunction of the diabatic matrix H

by the expression inside the square brackets in the left-hand side of Eq. (87) with the EI term omitted), of the square of the total nuclear orbital angular

^						^		^
momentum J, of its space-fixed z-component						Jz	and of the inversion operator I
of the nuclei through their center of mass G according to the expressions
^ nu	v	d;JM	¼ Ev	d;JM
H
^2	v	d;JM				2	v	d;JM
J			¼ JðJ þ 1Þh					ð94Þ
^		d;JM	¼ Mhv		d;JM
Jzv
^ d;JM			¼ ð 1Þ v			d;JM
Iv

In these equations, J and M are quantum numbers associated with the angular

^2	^	parity
momentum operators J
	and Jz, respectively. The number ¼ 0, 1 is a d;JM

quantum number that specifies the symmetry or antisymmetry of the v column vector with respect to the inversion of the nuclei through G. Note that the same parity quantum number appears for wd1;JM and wd2;JM . Also, the same irreducible representation symbol in these two components of vd appears, which does not mean that these diabatic nuclear wave functions transform according to the irreducible representation . Its meaning instead is as follows. The electronuclear Hamiltonian of the system is invariant under the group of permutations of identical AlAnAk atoms. For A3 it is the P3 group, for A2B it is the P2 group and for three distinct atoms ABC it is the trivial identity group. As a result, the Oðr; RlÞ that appears in Eq. (56) must transform according to an irreducible representation of the corresponding permutation group. The superscript signifies that the transformation properties of vd;JM

are	such that when taken together with			the transformation properties				of
el;d	ðr; qlÞ, they make	O	ðr; RlÞ belong to				d;JM
wel;d				. The separate factors wi				and
ci	ðr; qlÞ do not individually belong to			but	that their product does. In
						d;JM	are single valued,
addition, it is important to stress that these diabatic v

that is, are unchanged under a pseudorotation [26]. This behavior is the opposite

quantum reaction dynamics for multiple electronic states 315

to that of the adiabatic vad;JM , which must change sign under such pseudorotations, due to the geometric-phase effect.

Let us now expand the two nuclear motion partial waves w1d;JM

and w2d;JM

according to the following vector equation:

wd;JM ;nl0 l0

ð

r;

l

Þ

!

1

d;JM ;nl0 l0

w2

ðr; lÞ

d;J n0

0

d;

ð2Þ

l

l

¼ r

5=2

J

1

n1l

b1;n1l ; l

ðr; rÞ 1;n1l ; l

ð l

; rÞ

ð95Þ

DM l ð lð

ÞÞ0 P

d;J nl0 l0

d;

2

1

@ Pn2l

b2;n2l ; l

ð Þ

; rÞ A

Xl

ðr; rÞ 2;n2l ; l ð l

where, ðl1Þ refers to the set of three Euler angles aIl, ðl2Þ refers to the set of two hyperangles y, fl and l 0 is the absolute magnitude of the quantum number for the projection of the total angular momentum onto the body-fixed GzIl axis. Furthermore, the DJM l ð ðl1ÞÞ are the parity-symmetrized Wigner rotation functions defined as [33]

(

2J þ 1

)

1=2

DJ

ð

ð1Þ

Þ ¼

M l

l

16p2

½

1

1

Þ

Jþ

d

;0

þ ð

l

&

hDMJ l ð lð1ÞÞ þ ð 1ÞJþ þ l DMJ ; l ð lð1ÞÞi

ð96Þ

where DJM l ð ðl1ÞÞ is a Wigner rotation function of the Euler angles ðl1Þ [89]. The symmetrized Wigner functions have been orthonormalized according to

ð DMJ0 0 0 l0 DMJ

J0 0M0 0

ð97Þ

l dt ¼ dJ M l

l

where dt is the volume element for the Euler angles.

d;

ð2Þ

d;

ð2Þ

In Eq. (95), 1;n1l ; l

ð l

; rÞ and 2;n2l ; l ð l

; rÞ are the diabatic 2D (in

y; fl) local hyperspherical surface

functions

(LHSFs)

that depend

parame-

the eigenfunctions of

a

diabatic

reference

trically on r and are defined as

^ l

. This Hamiltonian can be chosen to be block diagonal, that is,

hamiltonian hd

h^d l ðy; fl; rÞ ¼

2mr2

^o2ðy; flÞ þ cosl2y

0

1

1

4 2

h2

1

0

!

d

0

98

e11

ðy; fl; rÞ

e22d

y; fl; r

ð

Þ

þ

0

ð

Þ

316

aron kuppermann and ravinder abrol

or have the off-diagonal diabatic couplings built in, that is,

h^d l ðy; fl; rÞ ¼ 2mr2

^o2

ðy; flÞ þ cosl2 y

0

1

1

4 2

h2

1

0

!

d

d

99

þ

e11

ðy; fl; rÞ

e12

ðy; fl; rÞ

ð

Þ

e21d

ð

y; fl; r

Þ

e22d

ð

y; fl; r

Þ

d;

ð2Þ

d;

ð2Þ

In the

former case, 1;n1l ; l ð l

; rÞ

and 2;n2l ; l

ð l

; rÞ

are

solutions of

uncoupled second-order partial differential equations, whereas in the latter case they are solutions of coupled differential equations and therefore their calculation requires a larger computational effort than to obtain the former.

^ l

Since, however, the reference Hamiltonian hd is independent of the total energy E of the system, the LHSFs need to be evaluated only once whereas the resulting scattering equations given by Eq. (101) must be solved for many values of E. As the off-diagonal diabatic couplings are built into Eq. (99), a smaller number of the corresponding LHSFs will be needed for convergence of the solutions of the scattering equations, as opposed to the ones resulting from Eq. (98), which do not have this off-diagonal coupling built in. Given the fact that the computational effort for solving those scattering equations scales with the cube of the number of LHSFs used, it is desirable to use LHSFs obtained from Eq. (99) rather than Eq. (98).

With either of these diabatic reference Hamiltonians, the LHSFs satisfy the eigenvalue equation

h^d l ðy; fl; rÞ0

d;

ð

Þ 1

¼

d;

ð

Þ

d;

ð

Þ

1

d;

;

y; fl; r

0 d;

;

r

d;

;

y; fl; r

1;n

l

E1;n

l

1;n

l

@

1l

1l

1l

2;n2l ; l ðy; fl; rÞ A

@ E2;n2l ; l ðrÞ 2;n2l ; l ðy; fl; rÞ A

ð100Þ

The diabatic LHSFs are not allowed to diverge anywhere on the half-sphere of

fixed radius r. This boundary condition furnishes the quantum numbers n1l and
^ l	has two angular
n2l , each of which is 2D since the reference Hamiltonian hd

degrees of freedom. The superscripts n0l; 0l in Eq. (95), with n0l refering to the union of n01l and n02l , indicate that the number of linearly independent solutions of Eqs. (94) is equal to the number of diabatic LHSFs used in the expansions of Eq. (95).

In the

strong

interaction region, the

diabatic

eigenfunctions

d;

i;n

i

;

ðy; fl; rÞ;

¼

;

are

l

l

i

1

2

themselves

expanded

in a

direct

product

of

two

orthonormal

basis

[90],

f

l

gnifl

l

l sets

l

niyl

ðy; rÞ

and

ðfl; rÞ, where nil

ðniyl ; nifl Þ. Both fniyl and gnifl

are chosen to be simple linear combinations

of trignometric functions [33] such that the resulting diabatic nuclear wave

quantum reaction dynamics for multiple electronic states 317

functions transform under the operations of the permutation symmetry group of identical atoms as described after Eqs. (94). Equations (100) are then transformed into an algebraic eigenvalue eigenvector equation involving the coefficients of these expansions, which is solved numerically by linear algebra methods. In the weak interaction region, where the coordinates r; ol; gl of Eq. (77) are used, the diabatic LHSFs are eigenfunctions of the appropriate reference hamiltonian expressed in those coordinates [33,90] and are labelled

d;
i;nil ; l	ðol; gl; rÞ; i ¼ 1; 2. These new LHSFs are expanded in the direct

product of the associated Legendre functions of cosgl and at a set of functions of ol determined by the numerical solution of a one-dimensional (1D) eigenfunction equation in ol [33,90]. Once the diabatic LHSFs are known, they provide the basis of functions in terms of which the expansion in Eq. (95) is defined. The diabatic nuclear wave function vector of that equation is then inserted into the first equation of Eqs. (94). Use of the orthonormality of the symmetrized Wigner functions (Eq. (97)) and integration over the 2D diabatic LHSFs, yields a set of coupled hyperradial second-order ordinary differential equations (also

d;J n0

0

called coupled-channel equations)

in the

l

l

coefficients b1;n1l ; l

ðr; rÞ and

d;J n0

0

d;J n0

0

l

l

l

l

b2;n2l ; l

ðr; rÞ. Let us define the column vectors bi

ðr; rÞ (i ¼ 1; 2) as

the vectors whose elements are scanned by nil ; l considered as a single row

index.

Let us also define a matrix Bd;J

ðr

;

whose n0

; 0

column vector is obtained

d;J n0

0

rÞ

l

l

d;J n0

0

l

l

l

l

by stacking the vector b2

ðr; rÞ under the vector b1

ðr; rÞ. These

vectors, for different n0l; 0l, are then placed side-by-side thereby generating a square matrix Bd;J whose dimensions are the total number of LHSFs (channels) used. The coupled hyperradial equation satisfied by this matrix has the form

	h2 d2	ð101Þ
	2m I dr2 þ Vd;J ðr; rÞ Bd;J ðr; rÞ ¼ EBd;J ðr; rÞ

where Vd;J ðr; rÞ is the interaction potential matrix obtained by this derivation procedure and that encompasses edðrÞ:

	d;J			d;J			!
Vd;J ðr; rÞ ¼	V11		r; r	V12		r; r		ð102Þ
	Vd;J		ðr; rÞ	Vd;J		ðr; rÞ
	21		ð Þ	22		ð Þ

Its dimensions are those of Bd;J ðr; rÞ.

C.Propagation Scheme and Asymptotic Analysis

The strong and weak interaction regions of the internal configuration space is divided into a certain number of spherical hyperradial shells. The 2D diabatic

318	aron kuppermann and ravinder abrol

LHSFs are determined at the center r of each shell. These LHSFs are then used to obtain the coupling matrix Vd;J ðr; rÞ given in Eq. (102). The coupled hyperradial equations in Eq. (101) are transformed into the coupled first-order nonlinear Bessel–Ricatti logarithmic matrix differential equation

d;J

2m

dF

ðr; rÞ

þ ½

Fd;J

; 2

½

EI

Vd;J

;

0

ð

103

Þ

h2

dr

ðr rÞ& þ

ðr rÞ& ¼

where

F

d;J

d;J

d;J

1

ð104Þ

ðr; rÞ ¼ ½ðd=drÞB

ðr; rÞ&½B

ðr; rÞ&

is the logarithmic derivative matrix and associated with Bd;J . Equation (103) is integrated from the beginning of each sector to its end using a highly efficient fourth-order logarithmic-derivative method [87,88] , and matched smoothly from one shell to another.

By using this method, the Fd;J matrix is propagated from a very small value of r ¼ ro, where a WKB solution is applicable, through a value rs that separates the strong and weak interaction regions, to an asymptotic value r ¼ ra where the interactions between different arrangement channels l has become negligible. At this asymptotic ra, the diabatic vd;JM is transformed to its adiabatic representation using the ADT matrix and matched to the asymptotic atom–diatom wave functions. This asymptotic analysis furnishes the reactance matrix RJ and from it the scattering matrix SJ [91,92]. For total energies E at which no electronically excited states of the isolated atoms or diatomic molecules are open, the elements of the open parts of these matrices correspond to the ground electronic atom and diatom products only. This is done for all and both parities ( ¼ 0; 1) and for a sufficiently large number of values of J (i.e., of partial waves) for the resulting differential and integral cross-sections to be converged. This numerical procedure for the current two- electronic-state case is closely related to that for a single-electronic-state described in [33].

V.SUMMARY AND CONCLUSIONS

A general treatment of quantum reaction dynamics for multiple interacting electronic states is considered for a polyatomic system. In the adiabatic representation, the n-electronic-state nuclear motion Schro¨dinger equation is presented along with the structure of the firstand second-derivative nonadiabatic coupling matrices. In this representation, the geometric phase must be introduced separately and the presence of a gradient term introduces numerical inefficiencies for the solution of that Schro¨dinger equation, even if

quantum reaction dynamics for multiple electronic states 319

the nonadiabatic couplings do not display any singular behavior at the intersections of adjacent electronic states. This makes it desirable to go to a diabatic representation that incorporates automatically the geometric phase effect. In addition, appropriate boundary conditions can be chosen so as to impart desired properties on the diabatic version of the n-electronic-state nuclear motion Schro¨dinger equation. One such property is the minimization of the magnitude of that gradient term. If a complete (infinite) set of adiabatic electronic wave functions is used in a Born–Huang expansion of the system’s electronuclear wave function (which is not possible in practice), this term vanishes automatically. In practice, a finite number n of adiabatic states are included for the treatment of chemical reactions. For this case, the gradient term survives in the diabatic representation as a nonremovable derivative coupling term, which, however, does not diverge at conical intersection geometries. A general method is presented that minimizes this nonremovable coupling term over the entire internal nuclear configuration space, leading to an optimal diabatization. As a very good first approximation, this gradient term can be neglected in the diabatic nuclear motion Schro¨dinger equation. Since that nonremovable coupling is obtained as a part of the diabatization process, its effect on the scattering cross-sections can be studied subsequently by perturbative or other methods.

A reactive scattering formalism for a triatomic reaction on two interacting electronic states is also presented. This formalism is an extension of the timeindependent hyperspherical method [26,33] for one adiabatic electronic state. The extended formalism involves obtaining diabatic local hyperspherical surface functions (LHSFs) for each hyperradial shell. The partial wave diabatic nuclear wave functions are expanded in terms of these diabatic surface functions and the coefficients of the expansion propagated to an asymptotic value of the hyperradius, where the diabatic nuclear wave function is transformed to its adiabatic counterpart. An asymptotic analysis of the adiabatic nuclear wave function gives the partial wave scattering matrices needed to obtain the desired differential and integral cross-sections. A comparison of the cross-sections obtained using this two-electronic-state formalism with those obtained using only the adiabatic ground electronic state with the geometric phase included, should provide an estimation of the energy range for which the one-electronic- state BO approximation is valid.

Acknowledgment

This work was supported in part by NSF Grant CHE-98-10050.

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