On the Characterization of Three-State Conical Intersections Using a

J. Phys. Chem. B 2006, 110, 19031-19039

19031

On the Characterization of Three-State Conical Intersections Using a Group Homomorphism Approach: The Two-State Degeneracy Spaces† Michael S. Schuurman* and David R. Yarkony* Department of Chemistry, Johns Hopkins UniVersity, Baltimore, Maryland 21218 ReceiVed: February 2, 2006; In Final Form: April 17, 2006

Second-order degenerate perturbation theory, in conjunction with the group homomorphism method for describing a similarity transformation, are used to characterize the subspace of two-state conical intersections contained in the branching space of a three-state conical intersection. It is shown by explicit calculation, using the lowest three-state conical intersection of (CH)3N2, that a second-order treatment yields highly accurate absolute energies, even at significant distances from the reference point of three-state intersection. The excellent agreement between the second order and ab initio results depends on the average energy component, which is computed using 5 first-order terms and 15 second-order terms. The second-order absolute energy change over the range F ) 0.0-0.3 au, where F is the distance from the three-state conical intersection in the branching space coordinates, is approximately 6500 and 9500 cm-1 for the E1)2 and E2)3 seams, respectively, with the maximum ab initio energy deviation from degeneracy of 200 cm-1 occurring at F ) 0.3 au. The characteristic parameters gIJand hIJ are also predicted to great accuracy, even at large F, with the error growing to only 10-15% at F ) 0.3 au.

I. Introduction It has only recently been demonstrated that conical intersections of three states of the same symmetry, once ignored as too rare to be of practical consequence, exist at energies and in regions of nuclear coordinate space where their impact can be significant.1-5,20 While the branching space (in which the degeneracy is lifted in a linear manner) for two-state conical intersections is two-dimensional, it is five dimensional in the case of a three-state conical intersection. One consequence of this larger dimension is that this branching space is not free of degeneracies. It contains a pair of three-dimensional degeneracy spaces.6 In one of these degeneracy spaces the upper two states are degenerate while in the other subspace it is the lower two states that remain degenerate. There have been few studies of these degeneracy spaces, although their effect on nuclear dynamics is expected to be profound. These degeneracy spaces give rise to linked two-state intersections,7 of states (i,j) and (j,k), near an (i,j,k) three-state intersection. Linked intersections complicate the geometric phase effect and can result in double-valued derivative couplings. The degeneracy space and its complications were originally studied by Keating and Mead6 and more recently by Han and Yarkony.8 In the first in a series of three studies dealing with conical intersections of three states, we developed a quasi-analytic description of a general accidental three-state conical intersection9 obtained from second-order perturbation theory using a group homomorphism approach.10 In this work, the second in the series, we show how perturbation theory and the group homomorphism approach provide both pedagogical insights and computational advantages in the study of the degeneracy subspaces of the branching space. Previous studies of the degeneracy spaces6,8 have been limited to a first-order descrip†

Part of the special issue “Robert J. Silbey Festschrift”. * Corresponding Authors. E-mail: [email protected]; [email protected].

tion of the vicinity of the conical intersection. In this work, the more robust second-order treatment is shown to have a dramatic effect on the prediction of the seam of two state intersection. Section II outlines the group homomorphism approach, explaining how it pertains to the determination of the degeneracy subspaces. Section III illustrates the ideas discussed in section II using the lowest energy three-state conical intersection in pyrazolyl, (CH)3N2.1 Section IV both summarizes the important results of this work and outlines future studies in which this methodology will be used to characterize the Nint - 5 dimensional seam space in its full dimensionality. II. Theoretical Approach The group homomorphism approach provides a means for dealing with similarity transformations of a matrix M using a set of basis matrices B(k). The resulting equation, referred to as the group homomorphism relation, can be used to advantage in locating and characterizing conical intersections of two states in the degeneracy space. (A) Basis Matrices. As shown in Appendix A, the following six matrices can be used as a basis for a 3 × 3 symmetric matrix, M:

( ) ( ) ( ) ( ) ( ) ( )

0 1 0 0 0 1 0 0 0 B(1) ) 1 0 0 , B(2) ) 0 0 0 , B(3) ) 0 0 1 0 0 0 1 0 0 0 1 0 1 B(4) ) 1/2 0 0 1 0 B(6) ) 0 1 0 0

0 0 -1 0 0 -1 0 , B(5) ) 1/2 0 -1 0 , 0 0 0 0 2 0 0 1

10.1021/jp0607216 CCC: $33.50 © 2006 American Chemical Society Published on Web 06/14/2006

(1a)

(1b)

19032 J. Phys. Chem. B, Vol. 110, No. 38, 2006

Schuurman and Yarkony

5 TABLE 1: R(U)l,k where U†B(k)U ) ∑l)1 B(l)R(U)l,k

k)1

l)1 l)3 l)5

cθc2φc2ψ - 1/4(3 + c2θ)s2φs2ψ sφc2ψsθ + 1/2cφs2ψs2θ s2ψ(1 - c2θ)/2

l)1 l)3 l)5

1/2s2θs2φsψ - sθc2φcψ cθcψsφ + c2θcφsψ s2θsψ

l)1 l)3 l)5

1/2cψs2θs2φ + sθc2φsψ c2θcφcψ - sψsφcθ cψs2θ

l)1 l)3 l)5

[2cθsψcψc2φ) + (1 + cθcθ)cφsφc2ψ]/2 [sθs2ψsφ - c2ψsθcθcφ]/2 -1/2sθsθc2ψ

l)1 l)3 l)5

-3/2sθcφsθsφ -3/2cθsθcφ cθcθ - 1/2sθsθ

k)2

k)3

k)4

k)5

The corresponding M[m] are given in terms of matrix elements of M in eq 1c:

M[m] ≡ Mk,l ) Ml,k for (m;k,l) ) (1;1,2), (2;1,3), (3;2,3) M[4] ≡ ∆M1,2 ) M1,1 - M2,2 M[5] ≡ ∆M1,2,3 ) [-M1,1 - M2,2 + 2M3,3]/3 M[6] ≡

∑M1,2,3 ) [M1,1 + M2,2 + M3,3]/3

∑

B R(U)k′,k, k ) 1 - ({N,N} - 1) (2a) (k′)

k′)1

and from eq 2a,

U†MU

)M ˜ becomes {N,N}-1

M ˜ ≡ M[{N,N}]B

({N,N})

+

∑ k′,k)1

M[k]B(k′)R(U)k′,k

(2b)

-c2θsφsψ + cψcφcθ 2sθcψs2φ + s2θc2φsψ

l)2 l)4

-c2θsφcψ - sψcφcθ s2qc2fcy - 2sqsys2f

l)2 l)4

[sθs2ψcφ + cθsθsφc2ψ]/2 [c2ψc2φ(1 + cθcθ) - 2cθs2φs2ψ]/2

l)2 l)4

3/2cθsθsφ -3/2c2φsθsθ

where c ) cos, s ) sin. R(U) can be obtained by tedious but straightforward algebra and is given in Table 1. (C) Locating the Degeneracy Space. Equation 2b provides a method for directly locating points in the degeneracy space. Consider the similarity transform of a 3 × 3 Hamiltonian matrix H(Q), that is

H ˜ (Q; ψ, θ, φ) ) U(ψ, θ, φ)†H(Q)U(ψ, θ, φ)

R(UV) ) R(U)R(V)

(3)

Equation 3 is readily verified by repeated use of eq 2a.9,10 The form of R depends on the choice of B and the form of U. Choosing U

)(

(5)

Here H is a function of the internal coordinates Q. Using eq 2b 5

H ˜ ≡ H[6]B(6) +

∑H˜ [k]B(k)

(6a)

k)1

where 5

H[l](Q)R(U)k,l ∑ l)1

for k ) 1-5

(6b)

From eq 1a, it is apparent that for H ˜ to be diagonal, H ˜ [l](Q; ψ, θ, φ) (the coefficient of B(l)) for l ) 1, 2, and 3, must equal zero:

H ˜ [l](Q; ψ, θ, φ) ) 0, l ) 1-3

(7a)

From eq 1c, H ˜ will have two degenerate eigenvalues provided

Equation 2b is the lynchpin of the subsequent analysis. R, which maps the N × N orthogonal matrices U onto a subset of the {N} × {N} invertible matrices, is a group homomorphism:

( (

l)2 l)4

H ˜ [k](Q; ψ, θ, φ) ≡

{N, N} - 1

U B U)

cφc2ψsθ - 1/2sφs2ψs2θ -2cθc2ψs2φ - 1/2(c2θ + 3)s2ψc2φ

(1c)

The scheme for mapping two index quantities to a single index summarized above will be employed throughout this paper. (B) Similarity Transformations and the Group Homomorphism. Given that the B(k) are a basis and the first {N, N} - 1 matrices (where {K, L} ) K(K - 1)/2 + L for K g L and 0 otherwise) are traceless, the similarity transformation U†B(k)U can be written † (k)

l)2 l)4

)(

cψ sψ 0 1 0 0 cφ sφ 0 U(ψ, θ, φ) ) -sψ cψ 0 0 cθ sθ -sφ cφ 0 0 0 1 0 -sθ cθ 0 0 1

)

)

cψcφ-cθsφsψ cψsφ+cθcφsψ sψsθ ) -sψcφ-cθsφcψ -sψsφ+cθcφcψ cψsθ sθsφ -sθcφ cθ (4)

H ˜ [l](Q; ψ, θ, φ) ) 0, l ) 4

(7b)

At first glance this analysis may seem flawed, since with four equations and three angles it would appear that only one geometrical parameter need be varied to achieve a two-state conical intersection. However, this would contradict the noncrossing rule,11 which states that two coordinates need to be varied to achieve a two-state intersection. However, von Neumann and Wigner observed in their original derivation of the noncrossing rule11 that at a point of two-state degeneracy only two angles are independent. In particular, imposing the requirement implied by eq 7b renders the angle φ superfluous. Thus choosing φ ) 0, eqs 7a and 7b can be solved for θ, ψ and two coordinates Qi, Qj. However before we can do this, we must describe the form of H. (D) Three-State Hamiltonian. The Hamiltonian H provides an approximation to the following electronic structure problem.

Two-State Degeneracy Spaces

J. Phys. Chem. B, Vol. 110, No. 38, 2006 19033

The wave functions ΨN(q; Q) satisfy the electronic Schro¨dinger equation:

[He(q; Q) - EN(Q)]ΨN(q; Q) ) 0

(8a)

From eqs 11 and 12, one observes that the 10 characteristic parameters p ) (h(l), l ) 1-5, sk, k ) 1-5) together with the basic vectors w ˆ (k), k ) 1-5 and their dual space w k (k), k ) 1-5 completely describe the H(1).9 H(1-2) involves coordinates from both the branching and seam spaces and is given by

where q(Q) denotes the coordinates of the Nel electrons (Nnuc nuclei). The ΨN are approximated by an expansion in the configuration state function (CSF) basis:

∑

(13)

A[l](2) ) Ab,[l] + Abs,[l] + As,[l]

(14a)

where

NCSF

Ψk(q; Q) ≡

(1-2) (1) (2) h[l] ) h[l] + A[l] l ) 1-6

ckR(Q)ψR(q;

Q) for k ) 1 - N

CSF

(8b)

R)1

Ab,[l] )

where the ck(Q) satisfies

[HCSF(Q) - IEk(Q)]ck(Q) ) 0

b,[l] (i) (j) ai,j w w ∑ i,j

As,[l] )

(8c)

and HCSF(Q) is the electronic Hamiltonian in the CSF basis. In principle, the NCSF × NCSF Hamiltonian in eq 8c describes all the electronic states at all nuclear arrangements Q. It is replaced by H(Q) which describes only three states and then only in the vicinity of Qx, a point of three-state conical intersection. Using degenerate perturbation theory, H is obtained at two levels of approximation:9 H(1) and H(1-2), which are correct to first and second order, respectively, in δQ, where Q ) Qx + δQ. Thus, the eigenvalue problem in eq 8c may be written as

H(m)(Q)U(m)(Q) ) U(m)(Q)E(m)(Q) m ) 1, 1-2 (9a) with

While Q denotes 3Nnuc - 6 ) Nint internal coordinates, H(1) can be described in terms of only five intersection-adapted coordinates, w(i), i )1-5. These coordinates are constructed as follows. Define

(10)

and use eq 1c to define h[k], k ) 1-6 from hk,l. Then for each k ) 1-5, define the unit vector, w ˆ (k):

w ˆ ) h[k]/h where

h(k)

) ||h[k]|| and

w(k)

(k)

is the displacement along

h(k)w(k) ≡ h[k]‚δQ ≡ h[k](1)

(11a) w ˆ (k)

by

(11b)

Since the w ˆ (k) are not orthonormal, we also require the dual basis w k (k) defined by w ˆ (k)‚w k (j) ) δj,k in terms of which h(k) ) w k (k)‚h[k]. For k ) 6 5

h(1) [6] )

si(w ˆ (i)‚δQ) ∑ i)1

(12a)

where

k (k) sk ) h[6]‚w

bs,[l] (i) (j) ai,j z z ∑ i,j

(m) (m) E(m) i ) ξi + h[6]

E(0)(Qx) + E(1-2)(Q) (9b)

(k)

bs,[l] (i) (j) ai,j w z ∑ i,j

(12b)

We augment the branching space coordinates with a set of seam space coordinates, zˆ (k), k ) 1 - (Nint -5) which are mutually orthogonal and orthogonal to the branching space coordinates. Using these coordinates, a point Q will be denoted Q ) (w, z) or Q ) w when all the z(k) ) 0.

(14b)

In this work we will be focusing on the branching space only, and thus only the first term in eq 14b is relevant here. However, that term involves 90 independent parameters since for each b,[l] ai,j 1 e j e i e 5 and l ) 1-6. More details concerning the Hamiltonian, determination of the parameters, and the relation of the coordinates w ˆ (k), k ) 1-5 to nuclear motion can be found in ref 9. From eq 6a, the first- and second-order total energies, Ei(1) and Ei(1-2), may be expressed as a sum of two terms: a state dependent component ξ(m) and a geometry dependent, but state independent, trace component:

E(Q) ) E(0)(Qx) + E(1)(Q) + E(2)(Q) )

hk,l(Q) ) ck(Qx)†∇HCSF(Q)cl(Qx) k, l ) 1-3

Abs,[l] )

(15)

where m ) 1, 1-2 in first and second order, respectively. (E) Hyperspherical Coordinates for Branching Space. It is convenient to replace the Cartesian branching space coordinates with hyperspherical coordinates. To define hyperspherical coordinates we introduce spherical polar coordinates q1, β, and R to replace w(2), w(3), and w(5) and polar coordinates q2 and γ to replace w(1) and w(4). Finally, the radii q1 and q2 are replaced by a hyperangle and hyperradius ω and F where

q1 ) F cos ω, q2 ) F sin ω

(16a)

w(1) ) F sin ω sinγ

(16b)

(2)

) F cos ω sin β cos R

(16c)

(3)

) F cos ω sin β sin R

(16d)

(4)

) F sin ω cos γ

(16e)

(5)

) F cos ω cos β

(16f)

w

w

w w

Note that internal coordinates defined in eqs 16a-16f differ slightly from those employed in ref 9. In the above definitions, the right-hand sides yield simply the branching space coordinate w(k), as opposed to the weighted coordinate, h(k)w(k), employed in ref 9. The immediate advantage of this definition is that F retains a geometric significance as the distance from the origin in the branching space. Employing eqs 11 and 12, the quantities (m) in eqs 16a-16f can be used to construct the H ˜ [l] (Q; ψ, θ, φ) m )1, 1-2 in eq 13. It will prove useful below to note that, in (m) hyperspherical coordinates, H ˜ [l] (Q; ψ, θ, φ)/F, m ) 1 is independent of F but for m ) 2 it is not. (F) Degeneracy Space. The degeneracy space is the subspace of the three-state branching space in which two states remain degenerate. The points of two-state degeneracy form two threedimensional manifolds: one where the upper two states are



TABLE 2: Degenerate Solutions of H(1)U(1)(θ, 0, ψ) ) E(1)U(1)(θ, 0 , ψ) H(1)

Ψi

Cartesian coordinates

hyperspherical coordinates

U(θi, 0, ψi)

H

Ψ1, Ψ2, Ψ3

w(1), w(2), w(3), w(4), w(5)

F, R, β, γ, ω

θ, ψ

H1,2 f -H1,2 H1,3 f -H1,3

-Ψ1, Ψ2, Ψ3

-w(1), -w(2), w(3), w(4), w(5)

F, 180 - R, β, -γ, ω

180 + θ, -ψ

H1,2 f -H1,2 H2,3 f -H2,3

Ψ1, -Ψ2, Ψ3

-w(1), w(2), -w(3), w(4), w(5)

F, -R, β, -γ, ω

180 - θ, -ψ

H1,3 f -H1,3 H2,3 f -H2,3

Ψ1, Ψ2, -Ψ3

w(1), -w(2), -w(3), w(4), w(5)

F, 180 + R, β, γ, ω

-θ, ψ

degenerate, and one where the lower two states are degenerate. These subspaces, in which the points are denoted Q+t and Q-t, respectively, emanate from the point Qx of three-state degeneracy. To our knowledge, the effect of second-order terms on the locus of these points has yet to be considered. The methodology developed in ref 9 to determine H(1-2) permits such an analysis. Comparing the predictions of H(1-2) to the results of ab initio calculations provides a stringent test of the accuracy of this approximation and an idea of the size of effects of second-order terms. In fact, we show by detailed computation that second-order effects can be quite large. (i) Locus of Points of Conical Intersection: Symmetries. The locus of points of conical intersection has been considered at first order in refs 6 and 8. The results through second order differ fundamentally from those at first order, in that the degeneracies and symmetries of H(1) are eliminated in H(1-2), as we now explain. Let the basis for H(1-2) (which is the same as that for H(1)) be denoted ψi(q; Q), j ) 1-3. The nature of these crude adiabatic basis functions is discussed in ref 9. Here it is sufficient to note that the spectrum of H(m), m ) 1, 1-2, is invariant to a change in the sign of one or more of the ψi(q; Q). The sign change ψi(q; Q) f -ψi(q; Q) changes H(m) i,k to (m) (1) Hi,k for k * i. For H the matrix elements are linear functions of the Cartesian coordinates. Therefore, equivalent changes in the signs of the H(m) i,k are readily induced by geometrical changes. Table 2 succinctly summarizes these relations. The first column gives the matrix elements whose signs change, by either changing the sign of the basis function (coordinate) in column two (three). The fourth column gives the R, β, and γ (which induce the changes in θ and ψ displayed in the fifth column) required to produce the results in column three. Columns one and two apply equally well to H(m), m ) 1, 1-2. From Table 1 we see for example that ψ3(q; Q) f -ψ3(m) (q; Q), changes the sign of H(m) 1,3 and H2,3 for m ) 1, 1-2. For H(1) this is equivalent to changing the sign of x and y which can be achieved by changing R f R + 180°; that is, the point w ) (w(1), w(2), w(3), w(4), w(5)) and the point w ) (w(1), -w(2), -w(3), w(4), w(5)) have the same spectrum at first order. Similar results are obtained for rows 1 and 2 in Table 1. Thus, each eigenstate of H(1) is 4-fold degenerate. When second-order effects are included this is no longer (or is only approximately) true, as the 4-fold degeneracy is lifted. In a similar fashion the degeneracy space for the lower two states can be used to locate the corresponding degeneracy space for the upper two states. If all the matrix elements were to change sign, a degeneracy of the lower two roots would be converted into a degeneracy of the upper two roots. For H(1) this is accomplished by the change of coordinates w f -w, which in turn is accomplished by the change of angles γ f 180° + γ, R f 180° + R, and β f 180° - β . Interestingly, ψ and θ are unchanged. Again, these results become approximations for H(1-2).

(ii) Locus of Points of Conical Intersection: Numerical Approach. In determining the degeneracy spaces, Q(t, three parameters are fixed and two are optimized. For H(1) the spectrum is independent of F. Therefore, when considering H(1) we fix two hyperangles and optimize the remaining two such that Q ∈ Q(t. This result is independent of F, which therefore becomes the third parameter. We will choose R and β as the variables to be optimized and γ and ω as fixed parameters. Therefore from the previous discussion for each choice of γ and ω we expect to find two sets of solutions, at R and R + 180°, which at first order are degenerate and independent of F and which at second-order become split and acquire a F dependence. (iii) Characteristic Parameters of the Conical Intersection. As described in earlier work,12 the linear part of a conical intersection of two states I and J, is described by four characteristic parameters p ) (g, h, s1, s2), where 2g and h are the norms of the energy difference gradient gIJ:

gIJ ) cI†(Q(t)∇HCSF(Q(t)cI(Q(t) cJ†(Q(t)∇HCSF(Q(t)cJ(Q(t) (17a) and interstate coupling gradient hIJ:

hIJ ) cI†(Q(t)∇HCSF(Q(t)cJ(Q(t)

(17b)

and s1 and s2, which describe the tilt of the principal axis of the double cone, are the projections of

2sIJ ) cI†(Q(t)∇HCSF(Q(t)cI(Q(t) + cJ†(Q(t)∇HCSF(Q(t)cJ(Q(t) (17c) along the directions gIJ and hIJ, respectively. These can be related to the 10 characteristic parameters of the three-state intersection as follows. Let Q-t ) Qx + λ-t be a point in the degeneracy space of the lower two states (equivalent results are obtained for Q+t). Define a new coordinate system centered at Q-t:

w′(k) + λ(k) ) w(k), k ) 1-5

(18)

Then the single index matrix elements of H(1-2) become (1-2) (1) (l) (λ + w′) ) h[l] (λ + w′(l)) + H[l]

(b,l) ai,j (w′(i) + λ(i))(w′(j) + λ(j)) ∑ i,j

(19a)

Let λ ≡ λ-t be a point of two-state conical intersection of H(m), where m ) 1, 1-2, provided (m) -t (λ ) ≡ H ˜ [l]

5

∑ H[k](m)(λ-t)R(U(m)(λ-t; θ, ψ))l,k ) 0, k)1 l ) 1-4 (19b)



Define (m) -t ∇H ˜ [l] (λ ) ≡

5

(m) -t [∇H[k] (λ )]R(U(m)(λ-t; θ, ψ))l,k ∑ k)1

(20a)

where

∂ ∂w

(i)

(1) -t H[k] (λ ) )

∂ (1-2) -t (1) (1) H[k] (λ ) ≡ δi,kh[k] +2 δi,kh[k] (i) ∂w

∑j ai,j(b,k)λ-t(j)

(20b)

Then from eqs 17a-17c: -t I,J,(m) -t ˜ (m) ≈ ∇H ˜ (m) 2gI,J,(m) ≈ ∇H [4] (λ ) h [1] (λ )

(21a)

Equation 21a is applicable to either m ) 1 or the more accurate m ) 1-2. However, from eq 20b the first order results are particularly simple. In that case gI,J,(1) and hI,J,(1) are five component vectors with components given by a h[k](1), a firstorder coupling matrix element at the origin, the point of threestate intersection, multiplied by an angular factor that reflects the mixings induced by the diagonalization, specifically:

Figure 1. Geometric parameters for the minimum energy 1,2,3 2A three-state conical intersectikon in pyrazolyl. All out-of-plane angles are zero. Bond lengths and bond angles are in Å and degrees, respectively.

-t (k) (1) -t (λ , θ, ψ))4,k and gwIJ,(1) (k) (λ ) ) h R(U -t (k) (1) -t hwIJ,(1) (λ , θ, ψ))1,k (k) (λ ) ) h R(U

(21b)

where the R are given in Table 1. III. Application: Three-State Conical Intersection in (CH)3N2 The results of Section II are now applied to the previously studied1 three-state conical intersection that arises from the three lowest 2A electronic states in pyrazolyl (C3H3N2). Particular attention will be paid to the effect of the inclusion of secondorder terms in the Hamiltonian. The ab initio electronic structure data presented in this study was obtained using a correlationconsistent polarized double-ζ (cc-pVDZ) basis set13 in conjunction with first-order multireference configuration interaction (MRCI)14-17 wave functions as described in ref 9. The geometric parameters of the minimum energy three-state conical intersection considered here are given in Figure 1. The first-order, charb,[l] coefficients acteristic parameters, p, and the second-order ai,j were taken from ref 9. All coordinate transformations and parameter optimizations were performed using the symbolic algebra package Mathematica,18 while the electronic structure computations were performed using the COLUMBUS program package.19 (A) Degeneracy Spaces. A section of the degeneracy spaces was determined as a function of F for 0.0 < F e 0.3 au with ω ) γ ) 45° and R and β optimized to determine a two-state conical intersection. Given these constraints, there will exist two solutions for both the 1,2 2A and 2,3 2A degeneracy spaces. In first order, these solutions correspond to the degenerate solutions (R, β, γ, ω) and (R + 180°, β, γ, ω), as shown in Table 2. However, when second-order terms are included, this degeneracy is lifted, and two unique solutions are observed. For example, for the 1,2 2A degeneracy, the values of (R, β) at F ) 0.1 au in Figure 2 are given by (29.9°, 105.5°), while the second solution (not pictured) occurs at (207.4°, 107.6°), corresponding to R2 ≈ 180° + R1, β2 ≈ β1. The 1,2 2A and 2,3 2A degeneracy spaces, plotted in Figures 2 and 3, respectively, were determined (1-2) , the second-order Hamusing eq 7 and employing H ˜ [l]

Figure 2. Energies of the three states comprise the three-state conical (1-2) intersection at F ) 0 at points in the E1)2 degeneracy space. Superscripts (1) and (1-2) refer to points computed at first- and secondorder perturbation theory, respectively, while (ab) corresponds to ab initio data.

iltonian. As these figures show, the optimized (R, β) yield rigorously degenerate second-order energies, labeled (1-2) (1-2) [PT2] in Figure 2 and E2)3 [PT2] in Figure 3. ThroughE1)2 out this discussion, [PT1] and [PT2] will refer to the molecular geometries predicted to be points of two-state intersection by the first- and second-order Hamiltonians, respectively. Thus, the first-order energies computed at the second-order optimized coordinates, labeled Ei(1)[PT2], will display a small but observable energy splitting. At second order, the (R, β) coordinates displayed a moderate F dependence, exhibiting a range of (30.2°, 107.7°) at F ) 0.001 au to (29.5°, 101.5°) at F ) 0.3 au for the 1,2 2A degeneracy and (107.1°, 61.5°) to (108.7°, 65.2°) for the 2,3 2A degeneracy in Figures 2 and 3, respectively. As stated in Section IIF, the first-order results are independent of F. Since the coordinate γ was confined to 45°, the 1,2 2A and 2,3 2A degeneracy spaces presented here represent unrelated solutions. However, as demonstrated in Section IIF and shown


Figure 3. Energies of the three states comprise the three-state conical (1-2) intersection at F ) 0 at points in the E2)3 degeneracy space. Superscripts (1) and (1-2) refer to points computed at first- and secondorder perturbation theory, respectively, while (ab) corresponds to ab initio data.

explicitly in Table 2, the 1,2 2A (2,3 2A) degeneracy space, with energies (E1, E1, E2) has a corresponding 2,3 2A (1,2 2A) degeneracy space with energies (-E1, -E1, -E2) obtained by simply taking (w(1), w(2), w(3), w(4), w(5)) f (-w(1), -w(2), -w(3), -w(4), -w(5)) at any point of two-state intersection. (B) Effect of Second-Order Terms. Figures 2 and 3 evince a significant second-order contribution to the energy, with the first-order result being qualitatively incorrect for F > 0.05 au while the second-order result is reliable for F e 0.3 au. The accuracy of the second-order results is further emphasized by noting the energy scale of Figures 2 and 3. The second-order perturbation theory results successfully reproduce the ab initio results 10 000 cm-1 above the three-state conical intersection. The source of this accuracy is further investigated by decomposing the first- and second-order energies into contributions from the trace component given by h(m) [6] (eqs 12a and 13) and the state-dependent term ξ(m) i , where m ) 1, 1-2. As shown in eq 15, the total energy is obtained by simply taking the sum of the two components. The trace contribution functions as a geometry dependent energy shift and is equal to the sum of the sk k ) 1-5 contributions at first order, while the second-order contribution includes Ab,[6] as well. This latter term, as Figures 4 and 5 illustrate, is responsible for the significant difference between the first- and second-order total energies. As both figures demonstrate, the energies ξi(1) and ξi(1-2) differ by a few hundred wavenumbers at F ) 0.3 au; however, at this same (1-2) point, h(1) differ by almost 9000 cm-1! [6] and h[6] An important measure of the theory’s success is the ability to reproduce points of conical intersection computed using ab initio methods. Figure 6 illustrates the deviation from degeneracy of the 1,2 2A and 2,3 2A seams using ab initio methods at geometries obtained from the first- and second-order Hamiltonians. Thus, at each value of F in Figure 6, the [PT1] and [PT2] results refer to computations at slightly different geometries, given that the first-order results are independent of F. The second-order results are particularly impressive, with the values (ab) -1 at F ) 0.22 of ∆E(ab) 12 [PT2] and ∆E23 [PT2] less than 100 cm au and displaying a maximum ∆E(ab) [PT2] of approximately IJ


(x) Figure 4. State dependent (ξ(x) i ) and average energy (h[6]) contributions to the total energy, where x ) 1 and 1-2 refer to first and through second order perturbation theory energies, respectively, for points in (1-2) the E1)2 degeneracy space.

(x) Figure 5. State dependent (ξ(x) i ) and average energy (h[6]) contributions to the total energy, where x ) 1 and 1-2 refer to first- and secondorder perturbation theory energies, respectively, for points in the (1-2) E2)3 degeneracy space.

(m) (m) 200 cm-1 at F ) 0.3 au, where ∆E(m) 12 ) E2 - E1 , m ) 1, 1-2, ab. However, the first-order results are only of qualitative accuracy and differ from the [PT2] results by a factor of 2, -1 and ∆ exhibiting energy splittings of ∆E(ab) 12 [PT1] ) 300 cm -1 at F ) 0.3 au. E(ab) [PT1] ) 500 cm 23 As stated above, the 4-fold degeneracy of the first-order degeneracy space solutions is lifted in second order. To illustrate this splitting, the full complement of solutions arising from a (1) (1) single E1)2 and E2)3 manifold are presented in Figures 7 and 8. At each value of F in both figures, each solution corresponds to a different geometry with all the structures approximately exhibiting the relations in Table 2. The second-order energies that appear in Figures 2 and 3 are denoted as solution “[1]” in both Figures 7 and 8.



Figure 6. Deviation from degeneracy of the ab initio energies for points in the first ([PT1]) and second ([PT2]) order degeneracy spaces.

Figure 8. Seams of two-state conical intersection determined at second (1) order arising from a single, four-fold degenerate E2)3 manifold.

Figure 7. Seams of two-state conical intersection determined at second (1) manifold. order arising from a single, four-fold degenerate E1)2

Figure 9. |gIJ,(x) × hIJ,(x)|, where x ) 1, 1-2, and ab refers to first order through second order and ab initio results, respectively, determined at points in the first ([PT1]) and second ([PT2]) order degeneracy spaces.

(C) Characteristic Parameters. At each point on the seam of conical intersection, it is possible to determine the characteristic parameters gIJ and hIJ. However, since gIJ and hIJ can only be made orthogonal via a rotation at a point of degeneracy, computing individual components of these vectors may prove problematic at points where EI ≈ EJ. In particular, the ab initio points, computed at geometries obtained from the second-order Hamiltonian, are only approximately degenerate (see Figure 6) and thus the electronic state rotation that produces orthogonal gIJ and hIJ (ref 10) will introduce error into each component of the vectors. To obtain a meaningful comparison between the ab initio and perturbation theory results, |gIJ × hIJ| (which is invariant to the rotation at a point of intersection) is reported in Figure 9 as a function of F. To test the sensitivity of |gIJ,(ab) × hIJ,(ab)|, the parameters R and β were optimized at F ) 0.15 au using ab initio data to determine a structure at which E(ab) and I E(ab) were degenerate to ∆E(ab) < 0.1 cm-1. The difference J IJ between the two values of |gIJ,(ab) × hIJ,(ab)|, as determined at

the structure predicted by second-order perturbation theory and that at the optimized structure, differed by 0.1% thereby confirming the invariance of this quantity with respect to small displacements from the seam of conical intersection. As predicted shown eq 20b, the first-order gIJ and hIJ vectors are independent of F and are thus constant for all points along the seam. The effect of second-order terms improves agreement with the ab initio results in Figure 9, although anecdotal evidence based on analysis of other seam solution sets for this system suggests that this is not always the case. However, again noting the scale of the plots, the percent error for the both the firstand second-order results remains small even out to F ) 0.3 au (this has been observed to be true for all solutions the authors have computed), displaying a maximum error less than 15% and 10% for the (I, J) ) (1, 2) and (2, 3) seams, respectively. IV. Summary and Conclusions This study represents the penultimate work in a three part series to elucidate the potential energy surfaces in the vicinity



of three-state conical intersection using degenerate perturbation theory. The first work in the series9 laid the groundwork by exploring the nature of the five-dimensional branching space at the intersection as well developing the tools to determine an approximately diabatic three-state Hamiltonian. In this report, a second-order Hamiltonian has for the first time been used to explicate the nature of the three-dimensional degeneracy subspace. The previously described 4-fold degenerate first-order solutions are split in second order, yielding multiple seams of two-state conical intersection from a single first-order manifold. The second-order energies compared very favorably to the ab initio results, displaying a maximum error at F ) 0.3 au of 10% and 2% in the absolute energy for points on the (I, J) ) (1, 2) and (2, 3) seams, respectively. Furthermore, the splitting of the ab initio results at F ) 0.3 au was only approximately 200 cm-1, a particularly good result considering the distance from the intersection. While the second-order total energies were quantitatively correct at all F considered in this study, the firstorder results quickly become qualitatively wrong for F > 0.1 au. However, the predominant source of this error lies not in the state-dependent part of the solution, but in the average energy component h(m) [6] m ) 1, 1-2. The ability of the perturbation theory results to accurately predict the magnitude of the gIJ and hIJ parameters at each point of two-state conical intersection has also been observed, with the maximum percent errors lying between 10 and 15%. The ability of these methods to reproduce ab initio data, even at large F, suggests that three-state Hamiltonian determined over the course of this series is ideal for pursuing dynamical studies, allowing the potential energy surface to be determined at minimal computational cost. The final work in this series will focus on the description of the Nint - 5 seam space in its full dimensionality. Particular emphasis will be placed on the efficient computation of the As,[l] and Abs,[l] polynomials shown in eq 14. Upon completion of this series, the theoretical framework that will exist for threestate conical intersections should be the equivalent to that of the more widely known two-state case. Acknowledgment. This paper is dedicated to Robert J. Silbey on the occasion of his 65th birthday. D.R.Y. is also pleased to acknowledge the support of NSF Grant CHE-0513952 Appendix A: Basis Matrices In this work we ultimately focus on 3 × 3 symmetric matrices, but to illustrate the generality of the approach we initially consider N × N symmetric matrices. (i) N × N Matrices. A general symmetric matrix M can be written as a linear combination of N(N + 1)/2-1 traceless symmetric matrices and the unit matrix I as follows: {N,N}-1

M ≡ M[{N,N}]I +

∑ k)1

M[k]B(k)

(A1)

where {K, L}) K(K - 1)/2 + L for K g L and 0 otherwise. We will choose the B(k) such that when k ){n, n} for some n, the matrix is purely diagonal, and purely off-diagonal otherwise. It is also important to distinguish between the single index quantity M[k], the matrix element Mk,l, and the kth component of a vector v, vk. Indeed, eq A1 provides a mapping from the two index matrix element Mi,j to the single index moiety M[k]. The mapping is made more precise below. The basis matrices will satisfy

N

B(i)‚B(j) )

∑ Bk,l(i) Bk,l(j) ) δi,j k,l)1

(A2)

so that M[k] ) B(k)‚M. Those matrices possessing off-diagonal elements only are defined by ({K,L}) ) Bk,l

δ{k,l},{ K,L} + δ{l,k},{K,L}

x2

(A3a)

Given this definition, we find that

M[{K,L}] ) x2MK,L, for N g K > L

(A3b)

The description of the diagonal elements is less straightforward. To construct the diagonal matrices, we start from the complete set of diagonal matrices D(n) k,l ) δk,lδn,l, n ) 1 - N and construct the manifestly independent set:

B[{N,N}] )

I

xN

so that M[{N,N}] )

1

xN

N

∑ Mm,m

(A4a,b)

m)1

B[{1,1}] ) 1 1 (δ1,1 - δ2,2) so that M[{1,1}] ) (M1,1 - M2,2), (A5a,b) x2 x2 and

Bk,l′({n,n}) ) δk,ld′l(n) for n ) 2 - (N - 1 )

(A6)

where d′l(n) ) [δn,l - N-1]. The Bk,l′({n,n}) ) δk,ld′l(n) do not satisfy eq A1. However, the ({n,n}) Bk,l ) δk,ld(n) l for n ) 2 - (N - 1 )

(A7)

do satisfy eq A1, where the d(n) are obtained by orthogonalizing the d′(n) to (1/x2, -1/x2, 0, ...., 0) and each other and normalizing the result. This completes the construction of the basis matrices. (ii) The 3 × 3 Case. For N ) 3, d(6) ) (1/x3, 1/x3, 1/x3), (1) d ) (1/x2, -1/x2, 0), and d′(3) ) (-1/3, 2/3, -1/3). d′(3) is orthogonal to d(6) as expected and Schmidt orthogonalizing d′(3) against d(1) gives d′′(3) ) d′(3) - (d(1)†‚d′(3))d(1) ) (-1/3, 2/3, -1/3) + 1/x2 (1/x2, -1/x2, 0) ) (1/6, 1/6, -1/3). Normalizing d′′(3) to obtain d(3), we find:

( ) ( )

1 1 0 x2 0 1 1 B(6) ) 0 x3 0

B(1) )

0 0 -1 0 0 0 0 0 1 0 0 1

B(3) )

1 x6

(

-1 0 0 0 -1 0 0 0 2

)

(A8)

Therefore, for N )3 we obtain eqs 1a-1c following the appropriate scaling and mapping of the above B(k) matrices, given by: (1/x2)B(1) f B(4), x2 B(2) f B(1), (x6/2)B(3) f B(5), x2 B(4) f B(2), x2 B(5) f B(3), x3 B(6) f B(6). References and Notes (1) (2) (3) (4) (5)

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Two-State Degeneracy Spaces (6) Keating, S. P.; Mead, C. A. J. Chem. Phys. 1985, 82, 5102. (7) Han, S.; Yarkony, D. R. J. Chem. Phys. 2003, 119, 5058. (8) Han, S.; Yarkony, D. R. J. Chem. Phys. 2003, 119, 11561. (9) Schuurman, M. S.; Yarkony, D. R. J. Chem. Phys. 2006, 124, 124109. (10) Han, S.; Yarkony, D. R. J. Chem. Phys. 2003, 118, 9952. (11) von Neumann, J.; Wigner, E. Phys. Z. 1929, 30, 467. (12) Yarkony, D. R. Acc. Chem. Res. 1998, 31, 511. (13) Dunning, T. H. J. Chem. Phys. 1989, 90, 1007. (14) Shavitt, I. The method of configuration interaction. In Modern Theoretical Chemistry; Schaefer, H. F., Ed.; Plenum Press: New York, 1976; Vol. 3, p 189. (15) Shavitt, I. The graphical unitary group approach and its application to direct configuration interaction calculations. In The Unitary Group for the EValuation of Electronic Energy Matrix Elements; Hinze, J., Ed.; Springer-Verlag: Berlin, 1981.

J. Phys. Chem. B, Vol. 110, No. 38, 2006 19039 (16) Paldus, J. In The Unitary Group for the EValuation of Electronic Energy Matrix Elements; Hinze, J., Ed.; Springer-Verlag: Berlin, 1981. (17) Shavitt, I. Unitary group approach to configuration interaction calculations of the electronic structure of atoms and molecules. In Mathematical Frontiers in Computational Chemical Physics; Truhlar, D. G., Ed.; Springer: New York, 1988. (18) Mathematica; Wolfram Research, Inc.: Champaign, IL, 2003. (19) Lischka, H.; Shepard, R.; Shavitt, I.; Pitzer, R.; Dallos, M.; Mu¨ller, T.; Szalay, P. G.; Brown, F. B.; Alhrichs, R.; Bo¨hm, H. J.; Chang, A.; Comeau, D. C.; Gdanitz, R.; Dachsel, H.; Erhard, C.; Ernzerhof, M.; Ho¨chtl, P.; Irle, S.; Kedziora, G.; Kovar, T.; Parasuk, V.; Pepper, M.; Scharf, P.; Schiffer, H.; Schindler, M.; Schu¨ler, M.; Zhao, J.-G. COLUMBUS, An ab initio Electronic Structure Program, 5.9 version, 2003. (20) Sarkar, B.; Adhikari, S. J. Chem. Phys. 2006, 124, 74101.

On the Characterization of Three-State Conical Intersections Using a

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