Page 1

Accurate Double Many-Body Expansion Potential Energy Surface for the Lowest Singlet

State of Methylene†

S. Joseph and A. J. C. Varandas*

Departamento de Quı ´mica, UniVersidade de Coimbra, 3004-535 Coimbra, Portugal

ReceiVed: December 2, 2008; ReVised Manuscript ReceiVed: January 16, 2009

A single-sheeted double many-body expansion potential energy surface is reported for ground-state CH2by

fitting accurate ab initio energies that have been semiempirically corrected by the double many-body-expansion

scaled-external-correlation method. The energies of about 2500 geometries have been calculated using the

multireference configuration interaction method, with the single diffusely corrected aug-cc-pVQZ basis set

of Dunning and the full valence complete active space wave function as reference. The topographical features

of the novel global potential energy surface are examined and compared with other potential energy surfaces.

1. Introduction

Methylene biradical, CH2, is of interest to both chemists and

astrophysicists. The electronic characteristics cause methylene

to possess rather unusual chemical and physical properties. The

ground state of methylene is a triplet state (X˜3B1) but methylene

has also a low-lying singlet excited state (a ˜1A1in C2Vsymmetry

or 11A′ in Cs). This appears as an electrophile and/or a

nucleophile, whereas the triplet electronic state possesses radical

properties. On the chemical side, methylene is a product of

photolysis of diazomethane (CH2N2) or ketene (CH2CO). In the

interstellar medium, CH2 plays an important role in carbon

chemistry. Although the molecule has been identified1in

interstellar space, it is the direct chemical precursor of the widely

observed CH radical as well as other carbon-bearing molecules.

In addition, CH2 is thought to be an important link in the

photodissociation sequence of cometary methane. Specifically,

CH4photodissociates into CH2, which in turn can photodisso-

ciate into CH + H or C + H2. Of all species in this sequence,

only CH has been observed. Studies on CH and CH4have been

done in detail, with the largest uncertainty lying on the CH2

photodissociation rate constant and branching ratio to CH + H

or C + H2.

Despite the great interest of the methylene radical in

astrophysics and the chemistry of hydrocarbons combustion,

there has been limited studies on the potential energy surface

(PES) of the lowest (first) singlet state of methylene. However,

the ground triplet state of methylene has been widely studied.

Theoretically, the most accurate calculations to date on the triplet

methylene PES have been conducted by Kalemos et al.2and

Comeau et al.3The former2examines four states (X˜3B1, a ˜1A1,

b˜1B1, c ˜1A1) of the CH2molecule using state-of-the-art ab initio

methods and basis sets and provides clues to understand both

the geometric and electronic structure of those states. The other

by Comeau et al.3has focused on the minimum to extract the

ro-vibrational energy levels of the ground and first excited states

of methylene radical. Concerning the global PESs involved in

the reaction C(3P) + H2(1Σg+) f CH(2Π) + H(2S), heretofore

labeled as R1, a three-dimensional (3D) PES has been built for

the triplet state of CH2by Knowles et al.4but their study did

not attempt to describe the conical intersections between3B1

and3A2PESs for geometries with C2Vsymmetry, both of3A′′

symmetry in Cs. They have reported a C2Vbarrier for C(3P) +

H2insertion of ∼70 kcal mol-1and a Csbarrier of ∼40 kcal

mol-1. The former result contrasts with the finding of Harding

et al.5who found a small barrier for the insertion reaction due

to the conical intersection between the two triplet surfaces. This

is responsible for changing the lowest energy mechanism for

reaction R1 from direct abstraction to insertion.

Concerning the reaction C(1D) + H2(1Σg+) f CH(2Π) +

H(2S) (R2), a 3D PES has been calculated by Whitlock et al.6

based on the valence-bond diatomics-in-molecules (VB-DIM)

method. The reaction was predicted to follow a C2Vinsertion

mechanism with no apparent barrier against a 9.7kcal mol-1

barrier for the collinear abstraction. Blint and Newton7estimated

a collinear barrier of 15kcal mol-1for abstraction. Later,

Bussery-Honvault et al.8computed an adiabatic global PES

(heretofore called BHL) for the first singlet state of CH2. Their

surface shows no barrier for the C2Vinsertion, while a barrier

of 12.35 kcal mol-1is present for the collinear reaction. This

value lies between the VB-DIM estimation of Whitlock et al.6

and the result of Blint and Newton.7

From the experimental side, a wide number of studies9–11have

led to the determination of an accurate equilibrium geometry

and a local analytical PES for the ground singlet state. However,

the wealth of papers concerning the singlet-triplet energy gap

shows that this splitting is subject to debate so that theses states

havebeenintensivelystudiedbothexperimentallyandtheoretically.

Dynamics calculations, using both the quasi-classical trajec-

tory (QCT) and quantum mechanical methods have recently

been carried out.12–20A more recent combined experimental and

theoretical study on the dynamics of the reaction C(1D) + H2

has also been reported21,22using the BHL8for the ground singlet

state PES. Unfortunately, visible discrepancies were found

between the theoretical calculations on the BHL8surface and

the experimental results. A possible explanation for these

discrepancies is that the 11A′8PES is not accurate enough in

the long-range regions.21,22Other explanations are possible

though, in particular due to the fact that there are five singlet

states that correlate with C(1D) + H2in addition to intersections

of the conical type involving different electronic states.23In other

words, nonadiabatic effects may play a significant role.

The purpose of this work is to investigate the major

topographical features of the ground singlet state potential

†Part of the “George C. Schatz Festschrift”.

* Corresponding author, varandas@qtvs1.qui.uc.pt.

J. Phys. Chem. A 2009, 113, 4175–4183

4175

10.1021/jp810600r CCC: $40.75

2009 American Chemical Society

Published on Web 03/18/2009

Page 2

energy surface of methylene by using DMBE24–28theory.

Specifically we will employ the single-sheeted DMBE formalism

which takes proper account of long-range forces. The paper is

organized as follows. Section 2 describes the ab initio calcula-

tions carried out in the present work. In section 3, we deal with

the analytical representation of the PES. While section 3.1

focuses on the two-body energy terms, section 3.2 concentrates

on the three-body ones. The main topographical features of the

DMBE potential energy surface are discussed in section 4. The

concluding remarks appear in section 5.

2. Ab Initio Calculations

The ab initio calculations have been carried out at the MRCI29

level using the full valence complete active space30(FVCAS)

wave function as reference. Such a FVCAS reference wave

function, which involves six correlated electrons in six active

orbitals (5a′ + 1a′′), amounts to a total of 50 configuration state

functions. For the basis set, we have selected the aug-cc-pVQZ

(heretofore denoted by AVQZ) of Dunning,31,32with the

calculations being carried out using the Molpro33package. About

2500 grid points were chosen to map the potential energy surface

of both the C-H2and H-CH regions. The former region is

defined by 1.0 e rH2/a0e 4.0, 1.2 e RC-H2/a0e 10.0 and 0.0

e γ/deg e 90 while, for the H-CH one, we have used a grid

defined by 1.5 e rCH/a0e 4.0, 1.0 e RH-CH/a0e 10.0, and 0.0

e γ/deg e 180. For both channels, r, R, and γ define the

associated atom-diatom Jacobi coordinates.

All calculated ab initio energies have been subsequently

corrected by the double many-body-expansion scaled-external-

correlation34(DMBE-SEC) method to account for excitations

beyond singles and doubles and, most importantly, for the

incompleteness of the basis set. The total DMBE-SEC interac-

tion energy is then written as

V(R))VFVCAS(R)+VSEC(R)

(1)

where

VFVCAS(R))∑

AB

VAB,FVCAS

(2)

(RAB)+VABC,FVCAS

(3)

(RAB,RBC,RAC) (2)

VSEC(R))∑

AB

VAB,SEC

(2)

(RAB)+VABC,SEC

(3)

(RAB,RBC,RAC)

(3)

and R ) {Ri} (i ) 1-3) defines the set of interatomic

coordinates. In turn, the first two terms of the SEC series

expansion assume the form

VAB,SEC

(2)

(RAB))VAB,FVCAS-CISD

(2)

(RAB)-VAB,FVCAS

FAB

(2)

(RAB)

(2)

(4)

VABC,SEC

VABC,FVCAS-CISD

(3)

(RAB,RBC,RAC))

(RAB,RBC,RAC)-VABC,FVCAS

(3)(3)

(RAB,RBC,RAC)

FABC

(3)

(5)

Following previous work,34the FAB

chosen to reproduce the bond dissociation energy of the

corresponding AB diatomic, while FABC

the average of the two-body F factors. Such a procedure leads

to the following values: FHH

FCHH

) 0.9323. Note that the calculated scaling factors are close

(2)parameter in eq 4 has been

(3)

has been estimated as

(2)) 0.9773 and FCH

(2)) 0.9098, and

(3)

to unit, which is accepted as a good requirement for the method

to yield accurate results.

3. Single-Sheeted Potential Energy Surface

Following early developments by Murrell and Carter,35we

have recently suggested36a switching function formalism to

approximate a multivalued potential energy surface by a single-

sheetedform.Forthetitlesystem,thespin-spatialWigner-Witmer

rules predict the following dissociation scheme

CH2(11A′)fH2(1Σg

+)+C(1D)

fH2(3Σu

fCH(2Π)+H(2S)

+)+C(3P)

(6)

Clearly, a proper description of the potential energy surface

would imply the use of a multisheeted formalism due to the

occurrence of two electronic states for the carbon atom. Since

we are mostly interested in the dissociation channel of CH2(11A′)

into H2(1Σg

H(2S), one must use a function that switches off the C(3P) state

in the C + H2dissociation channel. For this purpose, we have

suggested36the form

+) + C(1D), and CH(2Π) dissociates into C(3P) +

h(R1))∑

i)1

2

{1-tanh[Ri(R1-R1

i0)+?i(R1-R1

i1)3]} (7)

where R1represents the H-H distance, and Riand ?i(i ) 1, 2)

are parameters to be calibrated from the ab initio energies such

as to control the C(1D-3P) decay with increasing H-H distance

(see the left panel of Figure 1). A major advantage of this

switching formalism over the one originally proposed by Murrell

and Carter35is that it leads to a unique set of electronic states

at the three-atom asymptote.

In order to get a smooth three-body energy term, we multiply

h(R1) by an amplitude function that annihilates eq 7 at short-

range regions (small C-H2distances)

g(r1))1

2{1+tanh[R(r1-r1

0)]}

(8)

where r1is the distance of C to the center of mass of H2. The

parameters in g(r1) have been chosen so as to diminish any

abrupt switching behavior in the short-range part (see right-

hand-side panel of Figure 1). The complete switching function

has the form

f(R))g(r1)h(R1)

(9)

with the numerical values of the parameters in eqs 7-9 being

collected in Table 1.

Within DMBE theory, the total potential energy function is

expressed as

V(R))VC(1D)

(1)

f(R)+∑

i)1

3

[VEHF

(2)(Ri)+Vdc

(2)(Ri)]+[VEHF

(3)(R)+

Vdc

(3)(R)] (10)

where VC(1D)

between C(1D) and C(3P) electronic states, namely, VC(1D)

0.046604Eh. The following sections give the details of all other

analytical energy terms in eq 10.

3.1. Two-Body Energy Terms. The potential energy curves

of CH and H2 have been obtained using the extended

Hartree-Fock approximate correlation energy method for

diatomic molecules, including the united atom limit37(EHFACE2U)

(1)

is the energy (at the scaled AVQZ level) difference

(1)

)

4176

J. Phys. Chem. A, Vol. 113, No. 16, 2009

Joseph and Varandas

Page 3

which shows the correct behavior at the asymptotic limits. They

assume the general form37

V(R))VEHF(R)+Vdc(R)

(11)

where EHF refers to the extended Hartree-Fock type energy

and dc is the dynamical correlation energy. This is modeled

semiempiricaly as described38

Vdc(R))- ∑

n)6,8,10

where

?n(R))[1-exp(-AnR

In eq 13, Anand Bnare auxiliary functions24,25defined by An)

R0n-R1and Bn) ?0exp(-?1n) where R0, R1, ?0, and ?1are

universal dimensionless parameters for all isotropic interactions:

R0 ) 16.36606, R1 ) 0.70172, ?0 ) 17.19338, and ?1 )

0.09574. In turn, F ) 5.5 + 1.25R0is a scaling parameter, R0

) 2(〈rA2〉1/2+ 〈rB2〉1/2) is the LeRoy39parameter for onset the

undamped R-nexpansion, and 〈rX2〉 is the expectation value of

the square radius for the outermost electrons of atom X (X

) A, B).

The exponentially decaying part of the EHF type energy is

represented by the general form

Cn?n(R)R-n

(12)

F-BnR2

F2)]

n

(13)

VEHF(R))-D

R(1+∑

i)1

n

airi) exp(-γr)

(14)

where

γ)γ0[1+γ1tanh(γ2r)]

(15)

and r ) R - Reis the displacement from the equilibrium

diatomic geometry. In turn, D, ai, and γi (i ) 1,..., n) are

adjustable parameters to be obtained as described elsewhere.25,37

Here, we employed the accurate EHFACE2U potential energy

function for H2(X1Σg+) reported elsewhere.40To get the potential

energy curve of CH(2Π) in Figure,2 we used the same method,37

and those ab initio points calculated at the same level of theory

used for the title system. The numerical values of all parameters

for both diatomic potentials are collected in Table 1 of the

Supporting Information.

3.2. Three-Body Energy Terms. 3.2.1. Three-Body Dy-

namical Correlation Energy. The three-body dynamical cor-

relation energy assumes the usual form40

Vdc

(3))∑

i∑

n

fi(R)Cn

(i)(Ri,θi)?n(ri)ri

-n

(16)

where i labels the I-JK channel associated with the center of

mass separation and (ri, θi, Ri) are the Jacobi coordinates

corresponding to a specific R ) (R1, R2, R3) geometry of the

triatomic system (see Figure 1 of ref 26. Moreover, fi is a

switching function chosen from the requirement that its value

must be +1 at Ri ) Rieand ri f ∞ and 0 when Ri f ∞.

The three-body switching function assumes the form fi )

(1/2){1 - tanh[?(ηRi - Rj - Rk)]}, with corresponding

Figure 1. Switching function used to model the single-sheeted CH2DMBE potential energy surface. Shown in the left panel is the fit of the h(R1)

switching form to the ab initio points points calculated for C + H2as a function of H-H distance (R1). Shown in the right-hand side panel is a

perspective view of the global switching function in eq 9.

TABLE 1: Parameters in the Switching Function of

Equations 7-9

R1

R2

?1

?2

R110

R111

R120

R121

R

r10

0.571350

0.705610

1.001760

0.138958

2.315740

4.689200

3.846300

5.714090

0.75

5.5

Figure 2.

H2(1Σg+). The solid circles correspond to the ab initio points calculated

at the present level of theory.

EHFACE2U potential energy curves for CH(2Π) and

Potential Energy Surface of Methylene

J. Phys. Chem. A, Vol. 113, No. 16, 2009 4177

Page 4

expressions applying for fjand fk. Following a recent proposal41

to correct the long-range behavior, we adopt η ) 6 and ? to be

equal to 1.0a0-1. Regarding the damping function ?n(ri), we still

adopt eq 13 but replace R by the center-of-mass separation for

the relevant atom-diatom channel. In addition, the value of F

has been optimized by trial-and-error to get a good asymptotic

behavior of the dynamical correlation term, leading to F )

16.125a0.

The atom-diatom dispersion coefficients in eq 16, which are

given by

Cn

(i)(Ri))∑

L

Cn

L(R)PL(cos θi)

(17)

where PL(cos θi) denotes the Lth Legendre polynomial. The

expansion in eq 17 has been truncated by considering of only

the coefficients C60, C62, C80, C82, C84, and C100; all other

coefficients have been supposed to make insignificant contribu-

tions and hence neglected. To estimate the dispersion coefficients

we have employed the generalized Slater-Kirkwood formula,42

with the dipolar polarizabilities calculated in the present work

at the MRCI/AVQZ level of theory. As usual, the atom-diatom

dispersion coefficients so calculated for a set of internuclear

distances were fitted to the form

L,AC+DM(1+∑

Cn

L,A-BC(R))Cn

L,AB+Cn

i)1

3

airi)exp(-a1r-∑

i)2

3

biri)

(18)

where Cn

coefficients (see Table 2 of the Supporting Information; for L

* 0 Cn

been explained elsewhere.41The parameters that resulted from

such fits are also reported in Table 2 of the Supporting

Information, whereas the internuclear dependences of the

dispersion coefficients are shown in Figure 3. Note that in R )

0 the asymptotic limit for the isotropic component of the

dispersion coefficients are fixed in the value for the A-X

interaction, with X being the united atom of BC. As noted

elsewhere,40eq 16 causes an overestimation of the dynamical

correlation energy at the atom-diatom dissociation channels.

To correct such a behavior, we have multiplied the two-body

dynamical correlation energy for the ith pair by ∏j*i(1 - fj)

and correspondingly for the channels j and k. This ensures40

that the only two-body contribution at the ith channel is the

one of the JK diatom.

L, AB, for L ) 0, are the corresponding diatom dispersion

L, AB) 0). All parameters and coefficients in eq 18 have

3.2.2. Three-Body Extended Hartree-Fock Energy. By

removing, for a given triatomic geometry, the sum of the one-

body and two-body energy terms from the corresponding

DMBE-SEC interaction energies eq 1, which was defined with

respect to the infinitely separated ground-state atoms, one obtains

the total three-body energy. By further subtracting the three-

body dynamical correlation contribution eq 16 from the total

three-body energy that is calculated in that way, one obtains

the three-body extended Hartree-Fock energy. This will be

represented by the following three-body distributed-polynomial43

form

VEHF

(3))∑

j)1

7

Pj(Q1,Q2,Q3)∏

i)1

3

{1-tanh[γi

j(Ri-Ri

j,ref)]}

(19)

where Pj(Q1, Q2, Q3) is a sixth-order polynomial form in

symmetry coordinates which are defined as usual.44–46Figure 4

displays the reference geometries that define the displacement

coordinates used to write the EHF-type polynomials in eq 19.

As usual, we obtain the reference geometries Rij, refby first

assuming their values coincide with bond distances of the

TABLE 2: Stratified Root-Mean-Square Deviations (kcal

mol-1) of DMBE Potential Energy Surface

energy

Na

max devb

rmsd

N> rmsdc

10

20

30

40

60

80

100

140

200

250

1000

150

197

232

263

347

469

884

1875

2422

2451

2510

0.264

0.716

1.738

1.906

2.056

2.056

3.591

5.989

5.989

5.989

5.989

0.120

0.196

0.299

0.365

0.501

0.559

0.677

0.797

0.838

0.843

0.858

50

47

43

57

81

127

200

386

488

491

512

aNumber of calculated DMBE-SEC points up to the indicated

energy range.

cNumber of calculated DMBE-SEC points with an energy deviation

larger than the root-mean-square deviation (rmsd).

bMaximum deviation up to indicated energy range.

Figure 3. Dispersion coefficients for the atom-diatom asymptotic

channels of CH2as a function of the corresponding internuclear distance

of the diatom.

Figure 4. Reference geometries used in the present work for the three-

body EHF part of the potential energy surface. See tex for details.

4178

J. Phys. Chem. A, Vol. 113, No. 16, 2009

Joseph and Varandas

Page 5

associated stationary points. Subsequently, we relax this condi-

tion via a trial-and-error least-squares fitting procedure. Simi-

larly, the nonlinear range-determining parameters γijhave been

optimized in this way. The complete set of parameters amounts

to a total of 140 linear coefficients (ci; see the definition in the

Supporting Information), 21 nonlinear coefficients γij, and 7

reference geometries Rij, ref. Their optimal numerical values are

collected in Tables 3 and 4 of the Supporting Information. A

total of 2500 points has been used for the calibration procedure,

with the energies covering a range up to 500 kcal mol-1above

the CH2global minimum. Table 2 shows the stratified root-

mean-squared deviation (rmsd) values of the final potential

energy surface with respect to all the fitted ab initio energies.

As shown, the fit shows a maximum rmsd of 0.9 kcal mol-1up

to the highest repulsive energy stratum.

4. Features of DMBE Potential Energy Surface

We find no barrier for the C(1D) insertion into H2in the 11A′

PES of CH2, although for collinear geometries the DMBE

function predicts a barrier height of 13.2 kcal mol-1with respect

to the C(1D) + H2asymptote, which is located at R1 ) 1.85a0

and R2 ) 2.55a0. This result lies between the early theoretical

estimates of 9.76and 15.07kcal mol-1, while being in fairly

good agreement with the most recent theoretical predictions of

12.35 kcal mol-1(from the BHL8PES) and 12.3 kcal mol-1

(from the RKHS13PES).

The DMBE PES here reported shows a calculated well depth

of 180.25 kcal mol-1for the CH2complex, in its lowest singlet

state relative to the three atom dissociation limit, also in good

agreement with the best theoretical determination of 180.3 kcal

mol-1by Comeau et al.3Comparing to the BHL8PES calculated

with the MR-SDCI method and a LANO (local atomic natural

orbital) basis set, the DMBE PES predicts a well depth larger

by 2.05 kcal mol-1. Note that the calculated MRCI/AVQZ well

depth for the 11A′ state of CH2is 178.66 kcal mol-1, with the

Davidson correction (Q) leading to a further increase of 1.12

kcal mol-1. Note further that the scaling of the dynamical

correlation using the DMBE-SEC method leads to a well depth

of 180.25 kcal mol-1. Furthermore, we have performed com-

putationally expensive MRCI/AV5Z calculations for some

selected geometries, having obtained a well depth of 179.19

kcal mol-1. As seen, this is smaller by 0.59 kcal mol-1than the

MRCI(Q) value and 1.06 kcal mol-1than our estimate based

on DMBE-SEC. Since this accounts for the finiteness of both

the one electron basis set and CI expansion, it seems appropriate

that this method predicts a further lowering of 0.47 kcal mol-1

with respect to the Davidson (Q) correction. We find the

potential well in DMBE to be located at R ) 1.3a0and r )

3.26a0, which corresponds to a geometry of C2Vsymmetry with

R(C-H) )

2.09a0 and ∠HCH

results are in very good agreement with the available experi-

mental47data on CH2, R(C-H) ) 2.09a0and ∠HCH ) 102.4°.

Relative to the entrance channel, the calculated well depth of

CH2(100.01 kcal mol-1) in Table 3 is seen to be larger by about

10 kcal mol-1than the early valence-bond prediction of

Whitlock et al.6but in good agreement with the more recent

predictions of 99.8 kcal mol-1from BHL7and 99.75 kcal mol-1

from RKHS13PESs.

Table 3 also shows that the harmonic frequencies of the

DMBE PES are in good agreement with the calculated raw

MRCI/AVQZ values of 2915, 2990, and 1396 cm-1, with the

corresponding DMBE-SEC values being 2914, 2994, and 1393

cm-1. In fact, the deviations between the DMBE predicted

values and the MRCI ones are smaller (5 cm-1or less) than the

corresponding differences for the BHL and RKHS PESs.

The singlet state minimum is located slightly higher than the

triplet ground-state one. The separation between these two states

has been widely studied both theoretically48–51and experimental-

ly,3,52–54but the values obtained cover a wide range of energies55

(from 874.39 to 3427.61 cm-1) depending on the method

employed. We find the calculated MRCI/AVQZ value for T0,

)

102.4°. Clearly, these

TABLE 3: Stationary Point Properties of the CH2(11A′) PES

feature propertyab initioa

ab initiob

BHLc

RKHSd

DMBE others

C(1D) + H2

R1/a0

V/Eh

∆Ee

ω(H-H)/cm-1

R2/a0

V/Eh

∆Ee

ω(C-H)/cm-1

R1/a0

R2/a0

R3/a0

V/Eh

∆Ee

ω1(H-H)/cm-1

ω2(C-H)/cm-1

ω3(bend)/cm-1

R1/a0

R2/a0

R3/a0

V/Eh

∆Ee

ω1(C-H)/cm-1

ω2(C-H)/cm-1

ω3(bend)/cm-1

1.40 1.402

-0.1260(-0.1284l)

0.0

4399

2.12

-0.1320 (-0.1338l)

-3.76 (-3.38l)

2848

1.40 1.401.401

-0.12787(-0.1281m)

0.0

4396

2.12

-0.13373 (-0.1339m)

-3.68 (-3.61m)

2845

1.85

2.55

1.401f

0.0

4430

2.12

0.0

4420

2.12

0.0

4418

2.12

0.0

4403f

2.12g

-0.1337h

-3.80i, -3.73n(-3.83o)

2861g

CH + H

-3.82

2851

-3.86

2848

1.85

2.56

-3.85

2850

1.85

2.55

linear barrier

C-H-H

-0.1069

13.2

2624

1515

1232

3.2682

2.0972

2.0972

-0.28725

-100.01 (-99.75m)

2914

2994

1393

12.36

4565

1454

1117

3.26

2.09

2.09

12.37

4341

1454

1117

3.27

2.09

2.09

global minimum

3.27

2.09

2.09

3.2578

2.0957

2.0957

-0.28472

-99.5 (-99.2l)

2915

2990

1396

3.020j

2.097j

2.097j

0.2889k

-89.5i, -98.9n(-100.3o)

2806j

2865j

1352j

-99.70

2801

2955

1331

-99.8

2923

3136

1383

-99.75

2906

3069

1382

aReference 8.

gReference 59.hReference 68.iReference 6.jReference 47.kReference 3.lThis work, MRCI(Q).mThis work, DMBE-SEC.nReference 23

MRCI.oReference 23 MRCI(Q).

bThis work MRCI.

cReference 8.

dReference 13.

eRelative to the C(1D) + H2 asymptote (kcal mol-1).

fReference 58.

Potential Energy Surface of Methylene

J. Phys. Chem. A, Vol. 113, No. 16, 2009 4179

Page 6

3115.94 cm-1, to be in good agreement with the best experi-

mental determination of56,573147 ( 5 cm-1.

Figures 5-10 illustrate the major topographical features of

the CH2DMBE PES which has been calculated in the present

work. Clearly, it has a smooth and correct behavior over the

whole configuration space. Figure 6 presents the contour plot

of the PES for a fixed internal angle ∠HCH

shows that the exit channel presents no potential barrier toward

dissociation of CH2 into CH(2Π) + H(2S). Figures 7 and 8

represents the contour plots corresponding to the insertion and

abstraction of C(1D) + H2reaction. Specifically, Figure 8 shows

that the potential at collinear geometries rises rapidly up to a

barrier of 13.2 kcal mol-1. So, the reaction C(1D) + H2(1Σg+)

f CH(2Π) + H(2S) is expected to occur mainly via noncollinear

configurations. Figure 9 shows energy contours for a carbon

[C(1D)] atom moving around a ground-state hydrogen molecule,

while a corresponding plot for a hydrogen atom moving around

CH is presented in Figure 10. The salient features from these

plots are some of the most relevant stationary points for the

title system. A characterization of their attributes (geometry,

)

102.4°. It

energy, and vibrational frequencies) is reported in Table 3. Also

included for comparison are the results that have been obtained

from other PESs, ab initio calculations, and experimental

spectroscopic measurements.6,47,58,59

The range of barrier heights for linearity in the 11A′ state of

CH2obtained by ab initio calculation is pretty large,60varying

from 8000 to 10000 cm-1. For example, an early calculation61

using the rigid bender Hamiltonian gave a result of 10000 cm-1

while another one employing62the internally contracted MRCI

method using a FVCAS with an extended Gaussian basis set

and including also the Renner-Teller coupling effect gave a

result of 8800 cm-1. Furthermore, a high level ab initio

calculation employing3also the MRCI method with a ANO basis

Figure 5. Potential energy curve of the one body term [C(1D)-C(3P)]

referring to the excitation energy.

Figure 6. Contour plot for bond stretching H-C-H keeping the

included angle fixed at 102.4°. Contours are equally spaced by 0.02Eh

starting at -0.28Eh.

Figure 7. Contour plot for a C2Vinsertion of C atom into diatomic

H2. Contours are equally spaced by 0.006Eh, starting at -0.28Eh. The

dashed lies are contours equally spaced by 0.000075Eh, starting at

-0.127Eh.

Figure 8.

configuration. Contours are equally spaced by 0.00675Eh, starting at

-0.268Eh.

Contour plot for bond stretching C-H-H in linear

4180

J. Phys. Chem. A, Vol. 113, No. 16, 2009

Joseph and Varandas

Page 7

set but without inclusion of the Renner-Teller coupling gave

a barrier height of 9600 cm-1. In turn, the DMBE PES predicts

for the barrier to linearity in 11A′ CH2(see Figure 11) a height

of 9644 cm-1, a value in good agreement with the experimental

determination of 9800 cm-1by Duxbury and Jungen63obtained

by fitting a bending potential function to the (0, V2, 0) levels

for V2 ) 0-3 as observed in laser-induced fluorescence

(LIF)64,65and low-resolution dispersed fluorescence66experiments.

The isotropic and leading anisotropic potentials are two

important quantities for the study of scattering processes in the

11A′ CH2PES and are shown in Figure 12. The isotropic average

potential V0determines how close on average the atom and

molecule can approach each other, while the magnitude of V2

indicates whether or not the molecule prefers to orient its axis

along the direction of the incoming atom: a negative value

prefers the collinear approach while a positive one favors the

approach through an isosceles triangular geometry.

Finally, preliminary rate constant calculations have been

carried out for the reaction C(1D) + H2f CH + H by running

quasi-classical trajectories on the PES of the present work at a

temperature of 298 K. The calculations predict the rate constant

to be (1.34 ( 0.06) × 10-10cm3s-1when the vibrational and

Figure 9. Contour plot for an C atom moving around a fixed H2diatomic in equilibrium geometry R1 ) 1.401a0which lies along the X-axis with

the center of the bond fixed at the origin. Contours are equally spaced by 0.005Eh, starting at -0.17Eh. The dashed lies are contours equally spaced

by 0.000075Eh, starting at -0.128Eh.

Figure 10. Contour plot for an H atom moving around a fixed CH diatomic RCH ) 2.12a0, which lies along the X-axis with the center of the bond

fixed at the origin. Contours are equally spaced by 0.0075Eh, starting at -0.287Eh. The dashed lines are contours equally spaced by 0.000075Eh,

starting at 0.133Eh.

Potential Energy Surface of Methylene

J. Phys. Chem. A, Vol. 113, No. 16, 2009 4181

Page 8

rotational quantum numbers of H2are kept fixed at V ) 0 and

j ) 0. This shows good agreement with a recent state-specific

quantum dynamics result16on the RKHS13surface, (1.2 ( 0.6)

× 10-10cm3s-1, and the experimental67value of (2.0 ( 0.6) ×

10-10cm3s-1. The calculated thermalized rate constant is (1.09

( 0.06) × 10-10cm3s-1. Although only slightly off the

experimental error bar, we should recall that quantum effects

are ignored in the present estimate. A detailed account of the

ongoing dynamics calculations will be reported elsewhere.

5. Concluding Remarks

In this work, we have calculated a wealth of accurate MRCI

energies for the first singlet state of the methylene biradical,

which have subsequently been corrected using the DMBE-SEC

method and modeled analytically using DMBE theory. The

global DMBE PES so obtained shows good behavior over the

entire configuration space. Its various topographical features

have also been carefully examined and compared with the

corresponding attributes of other PESs, as well as experimental

data available in the literature. The good agreement observed

with the rate constant values reported in the literature recom-

mends it for future dynamics studies of the title reaction.

Acknowledgment. This work has the support of Fundacao

para a Ciencia e Tecnologia, Portugal: POCI/QUI/60501/2004

and REEQ/128/QUI/2005, under the auspices of POCI 2010 of

Quadro Comunita ´rio de Apoio III cofinanced by FEDER.

Supporting Information Available: Tables of numerical

values of all parameters for both diatomic potentials, parameters

in eq 18, and the three-body extended Hartree-Fock energy in

eq 19 and reference geometries of the extended Hartree-Fock

energy in Eq 19, and definition of the general three-body EHF

polynomial. This material is available free of charge via the

Internet at http://pubs.acs.org.

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