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

i

292	aron kuppermann and ravinder abrol

These coupling matrix elements are scalars due to the presence of the scalar Laplacian r2Rl in Eq. (25). These elements are, in general, complex but if we require the celi ;ad to be real they become real. The matrix Wð2ÞadðRlÞ, unlike its first-derivative counterpart, is neither skew-Hermitian nor skew-symmetric.

The wði;2jÞadðRlÞ are also singular at conical intersection geometries. The decomposition of the first-derivative coupling vector, discussed in Section II.C, also facilitates the removal of this singularity from the second-derivative couplings. Being scalars, the second-derivative couplings can be easily included in the scattering calculations without any additional computational effort. It is interesting to note that in a one-electronic-state BO approximation, the firstderivative coupling element wð11;1ÞadðRlÞ is rigorously zero (assuming real adiabatic electronic wave functions), but wð12;1ÞadðRlÞ is not and might be important to predict sensitive quantum phenomena like resonances that can be experimentally verified.

III.ADIABATIC-TO-DIABATIC TRANSFORMATION

	A. Electronically Diabatic Representation		ð	lÞ
ad	at the end of Section II.C, the presence of the	Wð1Þad
As mentioned				R

$Rl w ðRlÞ term in the n-adiabatic-electronic-state Schro¨dinger equation (15) introduces numerical inefficiencies in its solution, even if none of the elements of the Wð1ÞadðRlÞ matrix is singular.

This makes it desirable to define other representations in addition to the electronically adiabatic one [Eqs. (9)–(12)], in which the adiabatic electronic wave function basis set used in the Born–Huang expansion (12) is replaced by another basis set of functions of the electronic coordinates. Such a different electronic basis set can be chosen so as to minimize the above mentioned gradient term. This term can initially be neglected in the solution of the n-electronic-state nuclear motion Schro¨dinger equation and reintroduced later using perturbative or other methods, if desired. This new basis set of electronic wave functions can also be made to depend parametrically, like their adiabatic counterparts, on the internal nuclear coordinates ql that were defined after Eq. (8). This new electronic basis set is henceforth referred to as ‘‘diabatic’’ and, as is obvious, leads to an electronically diabatic representation that is not unique unlike the adiabatic one, which is unique by definition.

Let celn ;dðr; qlÞ refer to that alternate basis set. Assuming that it is complete in r and orthonormal in a manner similar to Eq. (10), we can use it to expand the total orbital wave function of Eq. (11) in the diabatic version of Born–Huang

expansion as	Z	widðRlÞciel;dðr; qlÞ	ð26Þ
Oðr; RlÞ ¼
X

quantum reaction dynamics for multiple electronic states 293

where, the celi ;dðr; qlÞ form a complete orthonormal basis set in the electronic coordinates and the expansion coeffecients wdi ðRlÞ are the diabatic nuclear wave functions.

As in Eq. (12), we also usually replace Eq. (26) by a truncated n-term version

Xn

Oðr; RlÞ widðRlÞciel;dðr; qlÞ	ð27Þ
i¼1

In the light of Eqs. (12) and (27), the diabatic electronic wave function column

vector wel;d

ð

r; ql

(with

elements ciel;d

ð

r; ql ;

i

¼

1; . . . ; n) is related

to the

elÞ;ad

ðr; ql

elÞ;ad

adiabatic one w

Þ (with elements ci

ðr; qlÞ; i ¼ 1; . . . ; n) by an n-

dimensional unitary transformation

w

el;d

ðr; qlÞ ¼

~

el;ad

ðr; qlÞ

ð28Þ

UðqlÞw

where

UyðqlÞUðqlÞ ¼ I

ð29Þ

UðqlÞ is referred to as an adiabatic-to-diabatic transformation (ADT) matrix. Its mathematical structure is discussed in detail in Section III.C. If the electronic wave functions in the adiabatic and diabatic representations are chosen to be real, as is normally the case, UðqlÞ is orthogonal and therefore has nðn 1Þ=2 independent elements (or degrees of freedom). This transformation matrix UðqlÞ can be chosen so as to yield a diabatic electronic basis set with desired properties, which can then be used to derive the diabatic nuclear motion Schro¨dinger equation. By using Eqs. (27) and (28) and the orthonormality of the diabatic and adiabatic electronic basis sets, we can relate the adiabatic and diabatic nuclear wave functions through the same n-dimensional unitary transformation matrix UðqlÞ according to

d	~	ad	ðRlÞ	ð30Þ
v	ðRlÞ ¼ UðqlÞv

In Eq. (30), vadðRlÞ and vdðRlÞ are the column vectors with elements wadi ðRlÞ and wdi ðRlÞ, respectively, where i ¼ 1; . . . ; n.

B.Diabatic Nuclear Motion Schro¨dinger Equation

We will assume for the moment that we know the ADT matrix of Eqs. (28) and (30) UðqlÞ, and hence have a completely determined electronically diabatic basis set wel;dðr; qlÞ. By replacing Eq. (27) into Eq. (13) and using Eqs. (7) and

(8) along with the orthonormality property of wel;dðr; qlÞ, we obtain for vdðRlÞ

294			aron kuppermann and ravinder abrol
the n-electronic-state diabatic nuclear motion Schro¨dinger equation
	h2		þ 2Wð1ÞdðRlÞ $Rl þ Wð2ÞdðRlÞg þ fedðqlÞ EIg vdðRlÞ
		fIrR2 l
	2m
¼ 0			ð31Þ

which is the diabatic counterpart of Eq. (15). The parameter edðqlÞ is an n n diabatic electronic energy matrix that in general is nondiagonal (unlike its adiabatic counterpart) and has elements defined by

d	el;d	^el	el;d	ðr; qlÞir	i; j ¼ 1; . . . ; n	ð32Þ
ei; j	ðqlÞ ¼ hci	ðr; qlÞjH	ðr; qlÞjcj

Wð1ÞdðRlÞ is an n n diabatic first-derivative coupling matrix with elements defined using the diabatic electronic basis set as

wið;1jÞdðRlÞ ¼ hciel;dðr; qlÞj$Rl cjel;dðr; qlÞir

i; j ¼ 1; . . . ; n ð33Þ

Requiring celi ;dðr; qlÞ to be real, the matrix Wð1ÞdðRlÞ becomes real and skewsymmetric ( just like its adiabatic counterpart) with diagonal elements equal to zero. Similarly, Wð2ÞdðRlÞ is an n n diabatic second-derivative coupling matrix with elements defined by

wið;2jÞdðRlÞ ¼ hciel;dðr; qlÞjrR2 l cjel;dðr; qlÞir

i; j ¼ 1; . . . ; n ð34Þ

An equivalent form of Eq. (31) can be obtained by inserting Eq. (30) into Eq. (15). Comparison of the result with Eq. (31) furnishes the following relations between the adiabatic and diabatic coupling matrices

Wð1ÞdðRlÞ ¼ U~ðqlÞ½$Rl UðqlÞ þ Wð1ÞadðRlÞUðqlÞ&	ð35Þ
Wð2ÞdðRlÞ ¼ U~ðqlÞ½rR2 l UðqlÞ þ 2Wð1ÞadðRlÞ $Rl UðqlÞ
þ Wð2ÞadðRlÞUðqlÞ&	ð36Þ

It also furnishes the following relation between the diagonal adiabatic energy matrix and the nondiagonal diabatic energy one

d	~	ad	ðqlÞUðqlÞ	ð37Þ
e	ðqlÞ ¼ UðqlÞe

It needs mentioning that the diabatic Schro¨dinger equation (31) also contains a gradient term Wð1ÞdðRlÞ $Rl wðRlÞ like its adiabatic counterpart [Eq. (15)].

quantum reaction dynamics for multiple electronic states 295

The presence of this term can also introduce numerical inefficiency problems in the solution of Eq. (31). Since the ADT matrix UðqlÞ is arbitrary, it can be chosen to make Eq. (31) have desirable properties that Eq. (15) does not possess. The parameter UðqlÞ can, for example, be chosen so as to automatically minimize Wð1ÞdðRlÞ relative to Wð1ÞadðRlÞ everywhere in internal nuclear configuration space and incorporate the effect of the geometric phase. Next, we will consider the structure of this ADT matrix for an n-electronic-state

problem and a general evaluation scheme that minimizes the magnitude of

Wð1ÞdðRlÞ.

C.Diabatization Matrix

In the n-electronic-state adiabatic representation involving real electronic wave

functions,

the skew-symmetric first-derivative

coupling

vector

matrix

Wð1Þad

R

lÞ

has n

ð

n

1

Þ

=2 independent nonzero coupling

vector

elements

wð1Þad ð

are

those

i; j

ðRlÞ;

ði ¼6 jÞ.

The ones having the largest magnitudes

1 ad

ðRlÞ

that

couple

adjacent adiabatic PESs, and therefore

the dominant wið; jÞ

are those for which j ¼ i 1, that is, lying along the

two off-diagonal lines

1

ad

ðRlÞ elements is

adjacent to the main diagonal of zeros. Each one of the wið; jÞ

associated with a scalar potential ai; jðRlÞ through their longitudinal component [see Eqs. (19) and (23)]. A convenient and general way of parametrizing the

1 ð

ql

Þ

n

n orthogonal ADT matrix U

of Eqs. (28) and (30) is as follows. Since

the coupling vector element wið; jÞ

ad

ðRlÞ couples the electronic states i and j, let

us define an

n n

orthogonal

i; j-diabatization matrix

[ui; jðqlÞ, with j > i]

k;l

k and

column l

element (k

;

¼ ;

2

; . . . ;

n) is designated by

whose row

l 1

ui; jðqlÞ and is defined in terms of a set of diabatization angles bi; jðqlÞ by the relations

uik;;jlðqlÞ ¼ cosbi; jðqlÞ	for k ¼ i and l ¼ i
¼ cosbi; jðqlÞ	for k ¼ j and l ¼ j
¼ sinbi; jðqlÞ	for k ¼ i and l ¼ j	38	Þ
¼ sinbi; jðqlÞ	ð
	for k ¼ j and l ¼ i
¼ 1	for k ¼ l ¼6 i or j
¼ 0	for the remaining k and l

This choice of elements for the ui; jðqlÞ matrix will diabatize the adiabatic electronic states i and j while leaving the remaining states unaltered.

As an example, in a four-electronic-state problem (n ¼ 4) consider the electronic states i ¼ 2 and j ¼ 4 along with the first-derivative coupling vector element wð21;4ÞadðRlÞ that couples those two states. The ADT matrix u2;4ðqlÞ can

296 aron kuppermann and ravinder abrol

then be expressed in terms of the corresponding diabatization angle b2;4ðqlÞ as

	0	1		0			0	0				1
u2;4 ql		0	cosb2;4ðqlÞ				0	sinb2;4ðqlÞ					39
ð	@	0		0			1	0				A	ð Þ
	B	0	sinb	2;4	ð	lÞ	0	cosb	2;4	ð	lÞ C
	Þ ¼ B				q					q		C

This diabatization matrix only mixes the adiabatic states 2 and 4 leaving the states 1 and 3 unchanged.

In the n-electronic-state case, nðn 1Þ=2 such matrices ui; jðqlÞ ðj > i with i ¼ 1; 2; . . . ; n 1 and j ¼ 2; . . . ; nÞ can be defined using Eq. (38). The full ADT matrix UðqlÞ is then defined as a product of these nðn 1Þ=2 matrices ui; jðqlÞ ð j > iÞ as

Y Y
n 1	n
UðqlÞ ¼	ui; jðqlÞ	ð40Þ

i¼1 j¼iþ1

which is the n-electronic-state version of the expression that has appeared earlier [72,73] for three electronic states. This UðqlÞ is orthogonal, as it is the product of orthogonal matrices. The matrices ui; jðqlÞ in Eq. (40) can be multiplied in any order without loss of generality. A different multiplication order leads to a different set of solutions for the diabatization angles bi; jðqlÞ. However, since the matrix UðqlÞ is a solution of a set of Poisson-type equations with fixed boundary conditions, as will be discussed next, it is uniquely determined and therefore independent of this choice of the order of multiplication, that is, all of these sets of bi; jðqlÞ give the same UðqlÞ [73]. Remembered, however, that these are purely formal considerations, since the existence of solutions of Eq. (44) presented next, requires the set of adiabatic electronic states to be complete; a truncated set no longer satisfies the conditions of Eq. (43) for the existence of solutions of Eq. (44). These formal considerations are nevertheless useful for the consideration of truncated Born–Huang expansion which follows Eq. (46).

We want to choose the ADT matrix UðqlÞ that either makes the diabatic firstderivative coupling vector matrix Wð1ÞdðRlÞ zero if possible or that minimizes its magnitude in such a way that the gradient term Wð1ÞdðRlÞ $Rl wdðRlÞ in

Eq. (31) can be neglected. By rewriting the relation between Wð1ÞdðRlÞ and Wð1ÞadðRlÞ of Eq. (35) as

Wð1ÞdðRlÞ ¼ U~ðqlÞ½$Rl UðqlÞ þ Wð1ÞadðRlÞUðqlÞ&

ð41Þ

quantum reaction dynamics for multiple electronic states 297

we see that all elements of the diabatic matrix Wð1ÞdðRlÞ will vanish if and only if all elements of the matrix inside the square brackets in the right-hand side of this equation are zero, that is,

$Rl UðqlÞ þ Wð1ÞadðRlÞUðqlÞ ¼ 0

ð42Þ

The structure of Wð1ÞadðRlÞ discussed at the beginning of this section, will reflect itself in some interrelations between the bi; jðqlÞ obtained by solving this equation. More importantly, this equation has a solution if and only if the elements of the matrix Wð1ÞadðRlÞ satisfy the following curl-condition [26,47,74–76] for all values of Rl:

curl wði;1jÞadðRlÞ&k;l ¼ ½wðk1ÞadðRlÞ; wðl1ÞadðRlÞ&i; j k; l ¼ 1; 2; . . . ; 3ðNnu 1Þ ð43Þ

In this equation, wðp1ÞadðRlÞ (with p ¼ k; l) is the n n matrix whose row i and column j element is the p Cartesian component of the wði;1jÞadðRlÞ vector, that is, ½wði;1jÞadðRlÞ&p, and the square bracket on its right-hand side is the commutator of the two matrices within. This condition is satisfied for an n n matrix Wð1ÞadðRlÞ when n samples the complete infinite set of adiabatic electronic states. In that case, we can rewrite Eq. (42) using the unitarity property [Eq. (29)] of UðqlÞ as

½$Rl UðqlÞ&U~ðqlÞ ¼ Wð1ÞadðRlÞ

ð44Þ

This matrix equation can be expressed in terms of individual matrix elements on both sides as

X

ð$Rl fi;k½bðqlÞ&Þfj;k½bðqlÞ& ¼ wði;1jÞadðRlÞ ð45Þ

k

where bðqlÞ ðb1;2ðqlÞ; . . . ; b1;nðqlÞ; b2;3ðqlÞ; . . . ; b2;nðqlÞ; . . . ; bn 1;nðqlÞÞ is a set of all unknown diabatization angles and fp;q½bðqlÞ& with p; q ¼ i; j; k are

matrix elements of the ADT matrix UðqlÞ, which are known trignometric functions of the unknown bðqlÞ due to Eqs. (38) and (40). Equation (45) are a set of coupled first-order partial differential equations in the unknown diabatization angles bi; jðqlÞ in terms of the known first-derivative coupling vector elements wði;1jÞadðRlÞ obtained from ab initio electronic structure calculations [43]. This set of coupled differential equations can be solved in principle with some appropriate choice of boundary conditions for the angles bi; jðqlÞ.

298	aron kuppermann and ravinder abrol

The ADT matrix UðqlÞ obtained in this way makes the diabatic firstderivative coupling matrix Wð1ÞdðRlÞ that appears in the diabatic Schro¨dinger equation (31) rigorously zero. It also leads to a diabatic electronic basis set that is independent of ql [76], which, in agreement with the present formal considerations, can only be a correct basis set if it is complete, that is, infinite. It can be proved using Eqs. (35), (36), and (42) that this choice of the ADT matrix also makes the diabatic second-derivative coupling matrix Wð2ÞdðRlÞ appearing in Eq. (31) equal to zero. As a result, when n samples the complete set of adiabatic electronic states, the corresponding diabatic nuclear motion Schro¨dinger equation (31) reduces to the simple form

	h2	ð46Þ
	2m IrR2 l þ fedðqlÞ EIg vdðRlÞ ¼ 0

where the only term that couples the diabatic nuclear wave functions vdðRlÞ is the diabatic energy matrix edðqlÞ.

The curl condition given by Eq. (43) is in general not satisfied by the n n matrix Wð1ÞadðRlÞ, if n does not span the full infinite basis set of adiabatic electronic states and is truncated to include only a finite small number of these states. This truncation is extremely convenient from a physical as well as computational point of view. In this case, since Eq. (42) does not have a solution, let us consider instead the equation obtained from it by replacing Wð1ÞadðRlÞ by its longitudinal part

$Rl UðqlÞ þ Wlonð1ÞadðRlÞUðqlÞ ¼ 0

ð47Þ

This equation does have a solution, because in view of Eq. (20) the curl condition of Eq. (43) is satisfied when Wð1ÞadðRlÞ is replaced by Wðlon1ÞadðRlÞ.

We can now rewrite Eq. (47) using the orthogonality of UðqlÞ as

~	ð1Þad	ðRlÞ	ð48Þ
½$Rl UðqlÞ&UðqlÞ ¼ Wlon

The quantity on the right-hand side of this equation is not completely specified since the decomposition of Wð1ÞadðRlÞ into its longitudinal and transverse parts given by Eq. (24) is not unique. By using that decomposition and the property of the transverse part Wðtra1ÞadðRlÞ given by Eq. (21), we see that

				$Rl Wlonð1ÞadðRlÞ ¼ $Rl Wð1ÞadðRlÞ							ð49Þ
and	since	Wð1Þad	ð	R	lÞ	is	assumed	to	have	been previously	calculated,
$Rl	1 ad
	Wlonð Þ	ðRlÞ is			known.		If we	take	the	divergence of both sides of

where fp;q

quantum reaction dynamics for multiple electronic states 299

Eq. (48), we obtain [using Eq. (49)]

½r2 ð Þ&~ð Þ þ ½$ ð Þ& ½$ ~ð Þ& ¼ $ ð1Þadð Þ ð Þ

Rl U ql U ql Rl U ql Rl U ql Rl W Rl 50

By using the parametrization of UðqlÞ this matrix equation can be expressed sides as

given by Eqs. (38) and (40) for a finite n, in terms of the matrix elements on both

Xk	½ðrR2 l fi;k½bðqlÞ&Þfj;k½bðqlÞ& þ ð$Rl fi;k½bðqlÞ&Þ ð$Rl fj;k½bðqlÞ&Þ&
¼ $Rl wið;1jÞadðRlÞ		ð51Þ

are the same as defined after Eq. (45). Equations (51) are a set of coupled Poisson-type equations in the unknown angles bi; jðqlÞ. For n ¼ 2, this becomes Eq. (68), as shown in Section III.D. The structure of this set of equations is again dependent on the order of multiplication of matrices ui; jðqlÞ in Eq. (40). Each choice of the order of multiplication will give a different set of bi; jðqlÞ as before but the same ADT matrix UðqlÞ after they are solved using the same set of boundary conditions.

By using the fact that for a finite number of adiabatic electronic states n, we choose a UðqlÞ that satisfies Eq. (47) [rather than Eq. (42) that has no solution], Eq. (35) now reduces to

1 d	~	ð1Þad	ðRlÞUðqlÞ	ð52Þ
Wð Þ	ðRlÞ ¼ UðqlÞWtra

This can be used to rewrite the diabatic nuclear motion Schro¨dinger equation for an incomplete set of n electronic states as

h2

1 ad

fIrR2 l þ 2U~ðqlÞWtrað Þ

ðRlÞUðqlÞ $Rl

þ Wð2ÞdðRlÞg

2m

þ fedðqlÞ EIg vdðRlÞ ¼ 0

ð53Þ

In

this

equation,

the

gradient term

~

Wð1Þad

ð

R

U

q

d

R

1 d

UðqlÞ

tra

lÞ

ð

lÞ $Rl v ð

lÞ ¼

Wð Þ

ðRlÞ $Rl vdðRlÞ

still appears

and,

as

mentioned

before,

introduces

numerical inefficiencies in its solution. Even though a truncated Born–Huang expansion was used to obtain Eq. (53), Wðtra1ÞadðRlÞ, although no longer zero, has no poles at conical intersection geometries [as opposed to the full Wð1ÞadðRlÞ matrix].

The set of coupled Poisson equations (50) can, in principle, be solved with any appropriate choice of boundary conditions for bi; jðqlÞ. There is one choice,

300	aron kuppermann and ravinder abrol
however,	for which bthe magnitude of Wtrað1ÞadðRlÞ is minimized. If at the

boundary surfaces Rl of the nuclear configuration space spanned by Rl (and the corresponding subset of boundary surfaces qbl in the internal configuration space spanned by ql), one imposes the following mixed Dirichlet–Neumann condition [based on Eq. (48)],

½$Rlb Uðqlb Þ&U~ðqlb Þ ¼ Wð1ÞadðRlb Þ

ð54Þ

it minimizes the average magnitude of the vector elements of the transverse coupling matrix Wðtra1ÞadðRlÞ over the entire internal nuclear configuration space as shown for the n ¼ 2 case [55] and hence the magnitude of the gradient term Wð1ÞdðRlÞ $Rl vdðRlÞ. To a first very good approximation, this term can be neglected in the diabatic Schro¨dinger Eq. (53) resulting in a simpler equation

	h2	ð55Þ
	2m fIrR2 l þ Wð2ÞdðRlÞg þ fedðqlÞ EIg vdðRlÞ ¼ 0

In this diabatic Schro¨dinger equation, the only terms that couple the nuclear wave functions wdi ðRlÞ are the elements of the Wð2ÞdðRlÞ and edðqlÞ matrices. The ðh2=2mÞWð2ÞdðRlÞ matrix does not have poles at conical intersection geometries [as opposed to Wð2ÞadðRlÞ] and furthermore it only appears as an additive term to the diabatic energy matrix edðqlÞ and does not increase the computational effort for the solution of Eq. (55). Since the neglected gradient term is expected to be small, it can be reintroduced as a first-order perturbation afterward, if desired.

In this section, it was shown how an optimal ADT matrix for an n-electronic- state problem can be obtained. In Section III.D, an application of the method outlined above to a two-state problem for the H3 system is described.

D. Application to Two Electronic States

In the two-electronic-state case (with real electronic wave functions as before), Eqs. (12) and (27) become

Oðr; RlÞ ¼ w1adðRlÞc1el;adðr; qlÞ þ w2adðRlÞc2el;adðr; qlÞ	ð56Þ
¼ w1dðRlÞc1el;dðr; qlÞ þ w2dðRlÞc2el;dðr; qlÞ	ð57Þ

Equations (28) and (30) are unchanged, with wel;dðr; qlÞ, wel;adðr; qlÞ, vdðRlÞ and vadðRlÞ now being two-dimensional (2D) column vectors, and Eq. (40) having the much simpler form

U b	q	cosbðqlÞ	sinbðqlÞ		ð	58	Þ
½ ð		lÞ& ¼ sinbðqlÞ	cosbðqlÞ

involving the single real diabatization or mixing angle bðqlÞ.

quantum reaction dynamics for multiple electronic states 301

Equations (31) and (32) are unchanged, with Wð1ÞdðRlÞ, Wð2ÞdðRlÞ, and edðqlÞ now being 2 2 matrices. The adiabatic-to-diabatic transformation, as for the n-state case, eliminates any poles in both the firstand second-derivative coupling matrices at conical intersection geometries but in this case Eq. (52) yields

Wð1Þd	R	Wð1Þad	ð	R	lÞ	ð	59	Þ
ð	lÞ ¼	tra

Elements of the matrix ðh2=2mÞWð2Þd are usually small in the vicinity of a conical intersection and can be added to ed to give a corrected diabatic energy matrix. As can be seen, whereas in Eq. (15) Wð1Þad contains both the singular

matrix Wðlon1Þad and the nonsingular one Wðtra1Þad, Eq. (31) contains only the latter. Nevertheless, the residual first-derivative coupling term Wðtra1Þad $Rl does not

vanish.

A ‘‘perfect’’ diabatic basis would be one for which the first-derivative coupling Wð1ÞdðRlÞ in Eq. (31) vanishes [10]. From the above mentioned considerations, we conclude, as is well known, that a ‘‘perfect’’ diabatic basis cannot exist for a polyatomic system (except when the complete infinite set of electronic adiabatic functions is included [26,47,74,76]), which means that Wð1ÞadðRlÞ cannot be ‘‘transformed away’’ to zero. Consequently, the longitudinal and transverse parts of the first-derivative coupling vector are referred to as removable and nonremovable parts, respectively. As mentioned in the introduction, a number of formulations of approximate or quasidiabatic (or ‘locally rigorous’) diabatic states [44,45,47–54] have been considered. Only very recently [77–82] have there been attempts to use high quality ab initio wave functions to evaluate the nonremovable part of the first-derivative coupling vector. In one such attempt [81], a quasidiabatic basis was reported for the triatomic HeH2 system by solving a 2D Poisson equation on the plane in 3D configuration space passing through the conical intersection configuration of smallest energy. It seems that no attempt has been made to get an optimal diabatization over the entire configuration space even for triatomic systems until now [55], aimed at facilitating accurate two-electronic-state scattering dynamics calculations for such systems. Conical intersections being omnipresent, such scattering calculations will permit a test of the validity of the one- electronic-state BO approximation as a function of energy in the presence of conical intersections, by comparing the results of these two kinds of calculations.

The ADT matrix for the lowest two electronic states of H3 has recently been obtained [55]. These states display a conical intersection at equilateral triangle geometries, but the GP effect can be easily built into the treatment of the reactive scattering equations. Since, for two electronic states, there is only one nonzero first-derivative coupling vector, wð11;2ÞadðRlÞ, we will refer to it in the rest of this