Accurate Explicit-Correlation-MRCI-Based DMBE Potential-Energy

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A: Molecular Structure, Quantum Chemistry, and General Theory

Accurate Explicit-Correlation-MRCI-Based DMBE Potential-Energy Surface For Ground-State CNO Cayo Emilio Monteiro Goncalves, Breno R. L. Galvão, Vinicius Cândido Mota, Joao Pedro Braga, and António J.C. Varandas J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.8b01881 • Publication Date (Web): 05 Apr 2018 Downloaded from http://pubs.acs.org on April 6, 2018

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The Journal of Physical Chemistry

Accurate Explicit-Correlation-MRCI-Based DMBE Potential-Energy Surface For Ground-State CNO C. E. M. Gon¸calvesa , B. R. L. Galvãob , V. C. Motac , J. P. Bragaa and A. J. C. Varandasd∗ a

Departamento de Qu´ımica, Universidade Federal de Minas Gerais, 31270-901, Belo Horizonte, Brazil

b

Departamento de Qu´ımica, Centro Federal de Educa¸cão Tecnol´ ogica de Minas Gerais, 30421-169, Belo Horizonte, Brazil

c

Departamento de F´ısica, Universidade Federal do Esp´ırito Santo, 29075-910, Vitória, Brazil d

Department of Physics, Qufu Normal University, China and

Department of Chemistry, and Chemistry Centre, University of Coimbra, 3004-535 Coimbra, Portugal E-mail: [email protected],[email protected]

1

ACS Paragon Plus Environment

The Journal of Physical Chemistry 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Abstract We report a new global double many-body expansion potential energy surface for the ground state of the CNO(2 A′ ) manifold, calculated by the explicit correlation multireference configuration interaction method. The functional form was accurately fitted to 3701 ab initio points with a root mean squared deviation of 0.99 kcal mol−1 . All stationary points reported in previous forms are systematically described and improved, in addition to three new ones and a characterization of an isomerization transition state between the CNO and NCO minima. The novel proposed form gives a realistic description of both short-range and long-range interactions, and hence is commended for dynamics studies.

1

Introduction

Despite considerable advances in atmospheric chemistry made by the scientific community, a vast number of different systems remains that require study, all with their own specificities at the molecular scale. Theoretical approaches can then play a prominent role, particularly in the case of small polyatomics where rather accurate methods can be used to simulate and reproduce the available experimental data. For such cases, the achieved accuracy depends critically on the availability of a reliable potential energy surface (PES), which accuracy is in turn determined both by the method used to calculate pointwise the energetic information and the quality of the functional form to which such data is fitted. For example, an accuracy close to the so-called chemical standard (1 kcal mol−1 ) can be reached whenever the system size is limited to only a few atoms. 1 In this regard, atmospheric chemistry is a convenient field where such theoretical studies can be applied since it involves mostly gas phase reactions between small molecules, with many such studies being available in the literature; 2–11 the list is by no means exhaustive, with the reader being addressed to the bibliographies of the cited papers from where other references may be obtained by cross-referencing. The CNO system has in particular received an increasing attention since it exhibits a rich 2


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chemistry. As we advance schematically in Figure 1 (further details will be examined later), two of its three possible reaction channels (C+NO and O+CN) are barrier-free and highly exothermic. 4,6,9 In the case of the C+NO channel, CN(X 2 Σ+ ) will be the dominant product since the C atoms are mostly attracted to the N valence electrons, forming a CNO* transition complex. As for CO(X 1 Σ+ ) formation, this is not expected to occur via a direct approach but rather via molecular rearrangement as predicted from transition state theory. 12 Moreover, because such reactions are quite exothermic, both N(X 4 S) and N(A2 D) atomic states will be produced. Note that ground-state CNO has three linear intermediates which are 2 Π Renner-Teller species: the 2 A′ and 2 A′′ surfaces become degenerate for linear arrangements.

C(3P)+ON(2+)

C(3P)+NO(2+)

SP3

O(3P)+CN(2+) CON

N(2D)+CO(1 +)

SP1 SP2

SP9

O(3P)+NC(2+)

N(2D)+OC(1

+)

CNO

NCO Figure 1: Schematic diagram illustrating the energy of the most important saddle points of the CNO PES and its three atom+diatom asymptotic limits.

Some important properties of the CNO system have already been measured through experiment, such as reactive cross sections and rate coefficients at low and room temper3



atures, rovibrational distributions, product energy partitioning, and electronic absorption spectra. 12–19 Theoretically, Halvick et al. initiated the study of this system with ab initio calculations of the CNO reaction pathways and its intermediates, and performed the first dynamics studies using an analytic representation of the ground state PES for quasiclassical trajectory (QCT) 20 calculations. In turn, Monnerville et al. 21 carried out quantum dynamics calculations of the collinear C+NO reaction to estimate the total rate coefficient using a capture approach through the long-range part of the potential, including spin-orbit coupling 22 effects. So far, the theoretical studies of the CNO system involved only the O+CN channel, since no information about the N+CO products had been obtained experimentally. The first attempt to build a global PES for the CNO system was due to Persson et al., 2 who employed the CASPT2 method to calculate the energetics of both 2 A′ and 2 A′′ adiabatic PESs, and which were subsequently fitted by Simonson et al. 3 to an analytic form. Geometries and vibrational frequencies were then reported for the minima and some transition states of both surfaces. The most recent PESs for this system were developed in a series of works by Nyman, Andersson, Markovic and Ambrahamsson, who have not only improved Persson’s PESs but also published new forms for both the 4 Σ− and two lowest 4 A′′ states. 4,5,7,9 They have extensively studied the dynamics of the C+NO and O+CN channels using QCT, having reported thermal rate coefficients in the temperature range of 300 − 4000 K. Such results were later improved by Frankcombe and Andersson 23 using quantum adiabatic dynamics. In this work, we present a realistic global PES for the 2 A′ state of the CNO system using the state-of-the-art explicitly correlated multi-reference configuration interaction (MRCIF12) 24,25 method, and the physically motivated double many-body expansion 26 (DMBE) theory for accurately fitting the energies so calculated. The various features of the present PES are compared with estimates previously reported in the literature and experimental properties of the system. The work is organized as follows. Section 2 describes the ab initio calculations, and section 3 the details of the DMBE modeling procedure. The results and main features of the PES are examined in section 4, while section 5 gathers the major

4


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conclusions.

2

Ab initio calculations

The PES uses a total of 3701 points, covering a grid of angles spaced by 15 degrees from 0 to 180, and bond distances 1.4 < RCO < 8.0 a0 , 1.3 < RNO < 10.0 a0 and 1.5 < RCN < 9.6 a0 ; see Figure 2 for the coordinates used in this work. The energy of each geometry was obtained by MRCI-F12 calculations, using the cc-pVQZ-F12 (VQZ-F12) basis set 27 and the MOLPRO 28 package for electronic structure calculations. The reference wave function was determined from complete-active-space-self-consistent-field (CASSCF) calculations, with 15 active electrons in 12 active orbitals. Suffice it to say that by considering explicitly the correlation between electrons inside the wave function, the MRCI-F12 method yields raw energies that approach the complete basis set (CBS) limit 24,25 better than other conventional MRCI approaches even when using some more extended basis sets. Thus, they are here assumed to lie sufficiently close to the CBS limit to allow their direct use (i.e., without CBS extrapolation 29 ) in modeling the PES up to an acceptable approximation. The three linear minima have also been optimized at the MRCI-F12/VQZ-F12 level using the quadratic steepest descent method. All energies were enhanced with the Davidson (+Q) correction, thus approximating up to quadruple excitations.

RCO

RNO

RCN Figure 2: Coordinates used in the present work. 5



3

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DMBE PES modeling

In DMBE theory, 26,30,31 a single-sheeted PES for a triatomic system such as the CNO is written as: V (R) = V (1) + V (2) (R) + V (3) (R)

(1)

where R = (RCO , RNO , RCN ) is a collective variable of the three diatomic internuclear separations. Normally, V (1) is set to the energy of the three separated atoms in their respective ground states (C(3 P ), O(3 P ) and N(4 S)). Because at the N+CO asymptotic channel the N atom is dissociated to its excited 2 D state, a new term is needed and V (1) is written as: V (1) = VC(3 P ) + VN(4 S),N(2 D) + VO(3 P )

(2)

where VC(3 P ) +VO(3 P ) can be set to zero and VN(4 S),N(2 D) is a function 10,32,33 defined to describe where appropriate the nitrogen in its excited state:

VN(2 D) = N

2

D −N

4

S

f (R)

(3)

with f (R) being a switching function, and N(2 D)−N(4 S) = 54.6178 kcal mol−1 (calculated at MRCI-F12(Q)/VQZ-F12 level). The switching function is defined as:

f (R) = g(JCO )h(RCO )

(4)

where JCO is N-CO Jacobi coordinate, g(JCO ) is written as 1 0 g(JCO ) = {1 + tanh y JCO − JCO } 2

(5)

and h(RCO ) is defined as h(RCO ) = {1 − tanh x1 (RCO − Ra ) + x2 (RCO − Rb )3 + x3 (RCO − Rc )5 } 6


(6)

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The data used to fit equation (6) is shown in figure 3. While the N atom is kept far from the CO, as the diatom bond increases the nitrogen switches from the A2 D state to the −3 −5 X 4 S. The optimum parameters are: x1 = 1.13391a−1 0 , x2 = 0.232454a0 , x3 = 0.35927a0 ,

Ra = 4.15981a0 , Rb = 4.95209a0 and Rc = 4.82206a0 . In turn, for g(JCO ), y = 0.8a−1 0 and 0 JCO = 6.0a0 , chosen conveniently to ensure a smooth insertion of N(2 D) into CO(1 Σ+ ).

60

50

relative energy / kcakmol−1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


40

30

20 ab initio fit 10

0 0

2

4

6

8

10

RC−O/a0

Figure 3: Fitting for h(RCO ), that describes the behavior of the N atom energy as the CO bond distance increases. The nitrogen is kept apart from the diatom by 30 bohr.

3.1

Two-body energy terms

The two-body potential energy, V (2) , is defined by the extended Hartree-Fock approximate correlation energy method, including the united atom limit (EHFACE2U), 34 which correctly describes the behavior at the asymptotes and has the general form: V (2) (R) = VEHF (R) + Vdc (R)

7


(7)


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where VEHF (R) refers to the extended Hartree-Fock energy (thence, the short-range part of the potential). In turn, Vdc (R) refers to the dynamical correlation energy, and hence may be linked to the long-range part of the potential. The exponential decay of the short-range part of the energy is represented by: D VEHF (R) = − R

1+

n X

!

ai δ i exp(−γδ)

i=1

(8)

where δ = R − Re is the displacement from the equilibrium diatomic geometry and γ = γ0 [1 + γ1 tanh(γ2 δ)]

(9)

The long-range contribution is written in a damped dispersion series: 35

Vdc (R) = −

X

Cn χn (R)R−n

(10)

n=6,8,10

where the damping functions χn (R) assume the form

χn (R) = 1 − exp

An R B n R 2 − ρ ρ2

n

(11)

An = α0 n−α1

(12)

Bn = β0 exp−β1 n

(13)

where ρ = 5.5 + 1.25R0 , and R0 is the LeRoy parameter defined elsewhere. 36 The parameters αi and βi are dimensionless and universal for all isotropic interactions: α0 = 16.36606, α1 = 0.70172, β0 = 17.19338 and β1 = 0.09574. The CO(X 1 Σ+ ), NO(X 2 Π) and CN(X 2 Σ+ ) potential curves were also modeled from MRCI-F12(Q)/VQZ-F12 ab initio energies. The parameters are gathered in Table S1 in the supplementary info.

8


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3.2

Three-body energy terms (3)

The three-body potential V (3) (R) is split into extended Hartree-Fock VEHF (R), dynamical (3)

(3)

correlation, Vdc (R), and electrostatic long-range, Velec (R), contributions plus a term specif(3)

ically modeled to describe any conical intersections, Vcusp (R). Thus: (3)

(3)

(3)

(3) V (3) (R) = VEHF (R) + Vcusp (R) + Vdc (R) + Velec (R)

3.2.1

(14)

Three-body dynamical correlation energy

The three-body dynamical correlation energy can be calculated from dispersion coefficients for the different atom-diatom subsystems. The analytical expression for this energy assumes the form 37 (3) Vdc (R)

=

3 X X i=1

fi (R)χn (ri )Cn(i) (Ri , θi )ri−n

(15)

n

where the index i represents the I–JK channel associated with the center of mass separation ~ i /|~ri R ~ i |; see Figure 1 of reference 37 for the ri ; Ri is the J–K bond distance, and cos θi = ~ri R notation. The function f (R) is a switching function which value must be +1 at Ri = Rie and ri → ∞, and 0 when Ri → ∞. One such form is 38 1 fi = {1 − tanh[ξ(ηsi − sj − sk )]} 2

(16)

where si = Ri − Rie (corresponding expressions apply for sj , sk , fj and fk ), and η and ξ are constants chosen in order to ensure the proper asymptotic behavior. In this work we have adopted ξ = 1.0 and η = 5.0 a−1 0 . Additionally, χn (ri ) is a damping function assuming the same form of equation11, with the difference that R is substituted for ri and the R0 = 9.5928 a0 , the average value for each of the three channels. (i)

Finally, in equation (15) the functions Cn (Ri , θi ) are atom-diatom dispersion coefficients

9



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which assume the form Cn(i) (Ri , θi ) =

X

CnL (Ri )PL (cos θi )

(17)

L

where PL (cos θi ) denotes the L-th term of the Legendre polynomial expansion. In the summation in equation (17) only the L = 0, 2 and 4 components have been considered, with the involved internuclear dependence being estimated as reported elsewhere, 39 i.e., using the dipolar polarizabilities calculated in the present work at MRCI-F12(Q)/VQZ-F12 level of theory together with the generalized Slater-Kirkwood approximation. 40 Atom-diatom dispersion coefficients were then fitted to the form CnA−BC (R) = CnAB + CnAC + DM

1+

3 X i=1

ai r i

!

exp −b1 r −

3 X i=2

bi r i

!

(18)

where r = R − RM is the displacement relative to the position of the maximum. The parameters that resulted from such fitting are reported in Table S5 of the SI, whereas the internuclear dependence of the dispersion coefficients is shown in panels a) to i) of figure 4. As noted elsewhere, 38 equation (15) causes an overestimation of the dynamical correlation energy at the atom-diatom dissociation channels. In order to correct such a behavior, we have multiplied the two-body dynamical correlation energy for the i-th pair by fi (R) and, correspondingly, for channels j and k. This ensures 38 that the only two-body contribution at the i-th channel is that of JK. 3.2.2

Electrostatic energy term

The electrostatic energy potential terms of the CNO system have their origin in the interaction of the quadrupole moments of the atoms C(3 P ), N(2 D) and O(3 P ) with the dipole and quadrupole moments of the diatomics CN(2 Π), CO(1 Σ) and NO(2 Π) respectively. Thence, similarly to the treatment in equation (15), the electrostatic energy of the CNO molecule

10


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9

b

c

6

5

C10/10 Eha0

-10

a

3

C010

C010

C010

0 2

d C08

1

3

C8/10 Eha0

-8

e C08

C28

C28

0

f

C08 C28

C48

C48

C48

-1 8

-6

g

C6/10Eha0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


C06

5

2

i

h C06

C26

C06

C26

C26

-1 0

3

6

9

0

3

6

9

0

R2/a0

R1/a0

3

6

9

R3/a0

Figure 4: Dispersion coefficients for atom-diatom asymptotic channels of CNO as a function of the corresponding atom-diatom separation. can be approximated by (3)

Velec (R) =

3 X

fi (R)[C4 (Ri , ri )ADQ (θa,i , γi , φab,i )ri−4 +

i=1

C5 (Ri , ri )AQQ (θa,i , γi , φab,i )ri−5 ]

(19)

where i, fi (R), Ri and ri have the same meaning as above, θa,i is the angle that defines the atomic quadrupole orientation, and φab,i is the corresponding dihedral angle and γ i is defined in Figure 2. The coefficients C4 (Ri , ri ) and C5 (Ri , ri ) for the i-th channel A − BC, are given

11



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by 3 C4 (Ri , ri ) = QA DBC (Ri )χ4 (ri ) 2 3 C5 (Ri , ri ) = QA QBC (Ri )χ5 (ri ) 2

(20)

where χ4 (ri ) and χ5 (ri ) are convenient damping functions of the type given by equation 11. DBC (Ri ) is the diatomic electric dipole moment, QBC (Ri ) is the corresponding diatomic electric quadrupole moment, and QA is the quadrupole moment of atom A. The angular variations of ADQ and AQQ are defined by 41 ADQ (θa , γ, φab ) = cos γ(3 cos2 θa − 1) + 2 sin θa sin γ cos θa cos γ cos φab

(21)

and AQQ (θa , γ, φab ) =1 − 5 cos2 θa − 5 cos2 γ + 17 cos2 θa cos2 γ + 2 sin2 θa sin2 γ cos2 φab + 16 sin θa sin γ cos θa cos γ cos φab

(22)

To eliminate the angle θa in equations (21) and (22) we use the classical optimizedquadrupole (COQ) model, 39,42–44 obtaining 45

θa = ∓ arctan{2 sin γ(8C5 cos γ + C4 R)/ {[256C52 sin2 γ cos2 γ + 2C4 C5 R cos γ(36 − 25 sin2 γ)+ 361C52 sin4 γ + 9C42 R2 + 144C52 ]1/2 − (5C42 R2 + 456C52 sin2 γ ∓ 3C 4 R cos γ ± C5 (19 sin2 γ − 12) + δ}}

(23)

which is used to replace θa in equations (21) and (22). A numerical parameter δ = 10−10 was included in this expression to prevent division by zero, and the solution corresponding to the equilibrium geometry of the diatomic molecules has been used. 12


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In order to model the variation of the CN(2 Π), CO(1 Σ) and NO(2 Π) quadrupole and dipole moments with the internuclear distance in equation (20) we have calculated those quantities at the MRCI-F12/VQZ-F12 level of theory as a function of the internuclear distance of the diatomic molecules. The quadrupole moment values so obtained are then fitted to the form 44

Q(R) = A 1 +

3 X i=1

ai r i

!

exp −

3 X i=1

bi r i

!

+

M6 χ8 (R) + Q∞ R6

(24)

where r = R − Rref and Rref is a reference distance corresponding to the maximum in the Q(R) curve, while the dipole moment values obtained are fitted to 44

D(R) = A 1 +

3 X

ai r i

i=1

!

exp −

3 X

bi r i

i=1

!

(25)

where r = R − Rref and Rref is a reference distance corresponding to the maximum in the D(R) curve. All the fitted parameters in equations (24) and (25) are given in Table S6 of the supporting information (SI), while a graphical view of the functions is shown in Figures 5a, 5b, 5c, 5d, 5e and 5f. Figures 5c and 5f also shows respectively the dipole and quadrupole moments of NO(2 Π) obtained previously 10 from MRCI(Q)/AVQZ calculations. As would be expected, the agreement with the current results is striking in the region populated by ab initio points, and the overall qualitative behavior is also very similar. 3.2.3

Three-body extended Hartree-Fock energy term

The extended Hartree-Fock part of the potential is described in polynomial form as follows: (3)

VEHF (R) =

m X i=1

Pij (Q1 , Q2 , Q3 ) ×

3 n Y

k=1

io (ref,i) 1 − tanh γki Rk − Rk h

!

(26)

where Pij (Q1 , Q2 , Q3 ) is an jth order polynomial that can be expanded as necessary and in this work we use j = 5. It is defined as: 13



9

a

b

c

d

e

f

D×10/ea0

6 3 0 -3 -6 5

3

Q/ea20

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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1 0

-1

-3 0

3

6

9 0

3

R1/a0

6

9

0

3

R2/a0

6

9

R3/a0

Figure 5: Dipole and quadrupole permanent electrostatic moments as a function of interatomic distance for CN(X2 Π), CO(X1 Σ) and NO(X2 Π); see figure 2 for the coordinates. Upper and lower panels show the results for the dipole and quadrupole moments, respectively. Open circles indicate ab initio moments calculated at MRCI-F12(Q)/VQZ-F12 level of theory, and solid lines the fits of equations (24) and (25). Indicated in dashed are MRCI(Q)/AVQZ results for the dipole and quadrupole moments from reference 10; see text.

a+b+c≤j

Pij (Q1 , Q2 , Q3 ) =

X

i Cabc Qa1 Qb2 Qc3

(27)

abc

The symmetry coordinates Q1 , Q2 and Q3 are defined as 

   p p p (ref,i) 1/3 1/3  RCO − R1  Q1   1/3       p  p (ref,i)  Q  =  0   1/2 − 1/2 R − R  2    NO  2   p  p p  (ref,i) Q3 2/3 − 1/6 − 1/6 RCN − R3 

Since each polynomial describes a part of the configuration space, several such polynomials may be necessary, and we use in this work m = 5. The problem to find the non-linear (ref,i)

parameters γki and Rk

is intricate since the the data to fit encompasses a variety of lo14


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(ref,i)

cal minima. Thence, we initially obtain the reference geometries Rk

of all stationary

points of the system and then determine by trial-and-error the γki parameters. All linear parameters of the polynomial are then fixed through a least-squares fitting procedure. Once a good combination of the non-linear parameters is obtained, we let all linear and non-linear parameters relax via a final global Levemberg-Maquardt fit to the whole set of data. When (3)

expanded to 5th order, each term of VEHF (R) has 56 linear and 6 non-linear parameters. 3.2.4

Conical intersection and its modeling

We have verified that there is a X 2 A′ /A2 A′ conical intersection for geometries where the CO bond is stretched, with an angle ∠CON of ∼ 120◦ . Such intersections are a common feature in molecular PESs, and cause cusps on the adiabatic energies that cannot be modeled with smooth (analytic) functions. Following pioneering work for the LiNaK system, 46 we have verified that the wavefunction changes sign when encircling a closed path around the crossing seam, as expected from Louguet-Higgins 47 sign-change theorems for a conical intersection (ab-c-d path in Figure 6). It has previously been shown 48–50 that such a cusp can be modeled without an adiabatic-to-diabatic transformation, provided that the crossing seam is fitted to a parametric equation (t, f (t), g(t)) in the space of internuclear distances {R1 , R2 , R3 }. For the present PES, we have found convenient to model the seam with a straight line, such that f (t) and g(t) are simply a + bt and c + dt, which is shown in figure 6 and yields the following set of parameters: a = 0.595908, b = 1.157979, c = 0.542636, and d = 1.867737. Briefly, at a given geometry {r1 , r2 , r3 } near the seam, the energy will depend linearly on its distance to the crossing line, which is obtained from ∆2 = [r1 − t0 ]2 + [r2 − f (t0 )]2 + [r3 − g(t0 )]2

(28)

where (t0 , f (t0 ), g(t0 )) is the closest point in the seam. To determine t0 , one then solves: d∆2 =0 dt 15


(29)


Page 16 of 31

Figure 6: CNO crossing seam in R-space. Ab initio degenerate geometries are plotted with solid black dots, and used to fit the parametric line shown. Using the above term, the following first-order polynomial is next constructed:

(3) Vcusp (R)

= T∆

a+b+c≤i X abc

cusp Cabc (RCO − Rc1 )a (RNO − Rc2 )b (RCN − Rc3 )c

(30)

where T is a range function chosen to ensure that the cusp term will not affect regions far from the crossing seam. The following Gaussian form centered at the crossing line has been found most convenient for our purposes: T = exp ζ2 [r2 − f (t0 )]2 + ζ3 [r3 − g(t0 )]2

(31)

with a suitable non-linear parameter for the present PES being ζ2 = ζ3 = 8, which has cusp been determined by trial and error. The values of the coefficients are C000 =-0.0270Eh , cusp cusp cusp C100 =0.0131Eh /a0 , C010 =0.0166Eh /a0 , C001 =-0.0010Eh /a0 , and the geometrical parame-

ters Rc1 =3.5222a0 , Rc2 =2.3748a0 and Rc3 =6.6632a0 . 16


Page 17 of 31 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


4

Results and discussion (3)

All VEHF (R) parameters values are listed in the tables S2-4 of the SI. In total, the global PES uses 571 parameters: 8 in V (1) , 14 in each diatomic of V (2) (R), 56 linear and 6 non-linear in (3)

(3)

(3)

(3)

each polynomial of VEHF (R), 13 in Vcusp (R), 126 in Vdc (R) and 72 in Velec (R). Table 1 lists the stratified root mean square deviation (RMSD) of the final fit, the maximum absolute error (max. dev.) and the RMSD per region of points with an specific energy above the global minimum NCO. The maximum RSMD is 0.99 kcal mol−1 which happens to be built up mostly from the highly repulsive regions of the PES. Table 1: Stratified root-mean-square deviations of the DMBE PES. Energya 10 15 20 25 50 75 100 150 200 250 300 >300

Nb 11 21 29 32 89 187 373 1983 3129 3416 3454 3701

max. dev.c 0.723 1.357 1.357 1.357 4.243 4.243 5.258 5.258 5.483 6.473 6.473 6.473

RMSD 0.463 0.543 0.576 0.567 0.762 0.767 0.901 0.654 0.835 0.883 0.889 0.992

d N>rmsd 4 8 12 13 20 51 75 443 756 802 814 779

a

All energies and deviations are in kcal mol−1 . b Number of points in the indicated energy range. Maximum deviation (absolute value of the absolute error) up to the indicated energy range. d Number of points with an deviation larger than the respective RMSD. c

Table 2 compares the properties of important minima and saddle points of the final PES with the MRCI(Q)-F12/VQZ-F12 ab initio results and the ones of the Andersson et al. 5 PES. The most important features are actually schematically depicted in the diagram shown earlier in Figure 1. As shown, the current PES describes accurately the geometries of all stationary points as directly obtained from the ab initio calculations (labeled M1-M3 and SP1-SP9): M1-M3 are the three linear minima of the system, with our results shown to be 17



in good agreement with the values reported in the literature, while the saddle points SP1SP9 will be discussed below. Not surprisingly, all our energies are about 15 kcal mol−1 lower in absolute value than the ones reported in Andersson’s work, which is due to the higher accuracy of the method here utilized. Yet, the energy differences between minima agree well between each other: 63.44 and 56.93 kcal mol−1 in our work versus 62.08 and 57.16 kcal mol−1 for Anderson’s PES, respectively for M1 − M2 and M2 − M3. In turn, the energy difference between the PES and the ab initio calculations at the minima is close to the RMSD of the PES itself, less than 1.5 kcal mol−1 . The only experimental geometry reported is from Misra et al. 51 for the NCO radical, with bond distances of RCN = 2.268 and RCO = 2.279 ±0.015 a0 . Our results, as the previous PES, gives an overestimation of the bond distances although a realistic comparison would clearly require some more recent experimental data. Figures 7 and 9 gather the major topographical features of the CNO PES and the most important saddle points. The C+NO channel is represented in figure 7 (a). The negative side of the abscissa (X axis) with Y=0 corresponds to the collinear approach of the C atom to NO from the N side (where M2 is visible), while in the positive side the approach is to the O atom (M3 being now visible). Both minima are deep, but the PES soon becomes repulsive when moving away from them, a trend actually visible for all three minima in all channels. Also visible are SP4 and SP7 when the C atom approaches almost perpendicularly, but both saddle points are of index 2 and of little importance in reaction dynamics. Only SP4 was predicted in the Andersson et al. 5 PES. In turn, SP3 is a linear transition state between CON and the C-NO asymptotic limit, lying 2.94 kcal mol−1 above the latter. This transition state was also predicted by Andersson with similar geometry and frequencies, but in their PES it lies less than 1 kcal mol−1 above the C-NO asymptote. Despite the small height of this barrier, it may have an important impact in low-energy collision dynamics studies. Figure 7 (b) shows the N+CO channel, in which the SP1 is also seen as a saddle point of index 2 between the CON and the N+CO, lying 12.77 kcal mol−1 above the asymptote, thence very similar to the one found in Andersson’s PES. Indeed, the good agreement extends

18


Page 18 of 31

Page 19 of 31 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Table 2: Energetics and structural properties of the present PES. RCN Linear minima NCO (M1) 2.308 2.311 (2.341) CNO (M2) 2.290 2.287 (2.319) CON (M3) 4.749 4.743 (4.764) Other important saddle points SP1 5.274 (5.389) SP2 2.306 (2.338) SP3 5.689 (5.835) SP4 4.013 (3.725) SP5 2.220 SP6 2.253 (2.281) SP7 7.241 SP8 7.069 SP9 5.918 a

RNO

RCO

E

ω1

ω2

ω3

ω4

4.539 4.536 (4.586) 2.293 2.297 (2.305) 2.506 2.505 (2.472)

2.231 2.225 (2.245) 4.583 4.584 (4.622) 2.243 2.238 (2.292)

-315.481 -314.294 (-299.303) -252.110 -251.842 (-237.224) -193.731 -194.060 (-180.057)

1986

1274

462

458

(1877) 1918

(1265) 1185

(518) 349

349

(1838) 1501

(1164) 635

(287) 253

253

(1455)

(932)

(476)

3.124 (3.197) 2.814 (2.793) 2.175 (2.203) 2.300 (2.328) 4.228 4.305 (4.256) 2.179 4.931 8.053

2.166 (2.192) 3.193 (3.241) 3.514 (3.632) 3.369 (3.233) 6.448 4.750 (4.586) 6.937 2.138 2.135

-190.714 (-172.308) -196.400 (-182.271) -148.602 (-139.309) -137.773 (-127.755) -180.602 -174.481 (-166.243) -151.102 -204.300 -203.382

1920 (1831) 1752 (1674) 1766 (1767) 1401 (1373) 1968 1881 (1986) 1894 8325 8034

402i (430i) 728 (739) 537i (393i) 733i (494i) 216i 348i (296i) 78i 872 1347i

80i (175) 952i (690i) 142 (139) 342i (354i) 132 242i (203i) 40i 223i 122

Energies are given in kcal mol−1 , frequencies (ω) in cm−1 , bond lengths in a0 . Data for the present 2 A′ PES, ab initio (bold) and previous PES of Andersson et al. 5 (in parentheses).

19


80i (175)

128 (139)

132

223i 122


to the values of the frequencies, but in his PES it is a true transition state. The SP8 and SP9 are shallow saddle points between the dissociated limit and the CON and NCO minima respectively, lying 1 kcal mol−1 below and 0.33 kcal mol−1 above the dissociation energy. A view of the O+CN channel is shown in Figure 7 (c), where SP2 is visible as a barrier for the NCO-CNO isomerization lying 119.15 and 55.71 kcal mol−1 above the NCO and the CNO minima (respectively), and 17.39 kcal mol−1 below the asymptotic limit. Andersson’s PES also predicts a similar transition state but little attention to it has been given. There is also a saddle point of index 2 (SP6) that offers a 4.52 kcal mol−1 barrier to the T-shaped approach of the O atom to the CN molecule. Also visible is SP5 which is only 1.60 kcal mol−1 below the asymptotic limit.

20


Page 20 of 31


10

8

8

6

6 y/bohr

10

4

2

4

2

0 -10

O -6

N

-2

2

6

0 -10

10

C -6

O

-2

2

x/bohr

6

10

x/bohr

(a)

(b) 10

8

6 y/bohr

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

y/bohr

Page 21 of 31

4

2

0 -10

C -6

N

-2

2

6

10

x/bohr

(c)

Figure 7: Partially relaxed contour plot for the C moving around NO (a), N around CO (b) and O around CN (c), with the diatoms lying along the x-axis and allowed to relax from 1.9 to 2.5 a0 . Energies start at −315.48 kcal mol−1 , with each of the 60 contours spaced by 5 kcal mol−1 . The 20 dashed contours start at −152, −205 and −180 kcal mol−1 for (a), (b) and (c) respectively, spaced by 0.15 kcal mol−1 .

The cusp found in the PES described by Eq. (30) can be seen in Figure 8, a contour plot for a fixed included angle of 120 degrees. The cusp arises close to RCO = 3.3a0 . Finally, Figure 9 shows all major features of the PES in a relaxed triangular plot using the scaled hyperspherical coordinates (β ∗ = β/Q and γ ∗ = γ/Q), thus allowing to establish a

21



connectivity between all stationary points in a more physical and multidimensional way. 52      2 1  RCN  Q 1 1    √   √   β  = 0  R2  3 − 3      CO       2 γ 2 −1 −1 RNO 5

4

RCO/a0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 22 of 31

O

3

2 2

3

4

5

RNO/a0

Figure 8: Contour plot for the PES for a fixed angle of 120 degrees in which the cusp can be seen at the red circle, constructed with 25 curves starting at −274 kcal mol−1 equally spaced by 10 kcal mol−1 .

22


Page 23 of 31

NCO 1

γ*

CN

N+ CO

O+

M1 M2 M3 SP1 SP2 SP3 SP4 SP5 SP6 SP7 SP8 SP9

0

N

CN

O

CO

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


-1

C+NO -1

0

1

β*

Figure 9: Relaxed triangular plot for the PES in hyperspherical coordinates 52 of the CNO PES, contours at −315 kcal mol−1 and equally spaced by 3.15 kcal mol−1 .

5

Conclusions

We performed extensive up to date explicitly correlated ab initio calculations to map the configuration space of CNO(X2 A′ ) and use the data so obtained to construct a single-sheeted DMBE PES which accurately fits all 3701 calculated points. All known stationary points were characterized, with the topographical features of the novel PES showing good agreement

23



with previous theoretical work and experimental data whenever available. An isomerization transition state between the CNO and NCO minima and two new saddle points of index 2 have also been identified and characterized. Given its high quality, the global PES here reported is therefore expected to be valuable for dynamics simulations and kinetics studies of reactions involving the ground state CNO system, work that is currently in progress.

6

Acknowledgments

The authors would like to thank CAPES, FAPEMIG, FAPES and CNPq for the financial support. The support of Edital 2015 do Programa institucional de Fundo de Apoio à Pesquisa da Universidade Federal do Esp´ırito Santo, as well as of the Laboratory of Computational Quantum Chemistry of the Departament of Physics of Universidade Federal do Esp´ırito Santo, are also acknowledged. One of us (AJCV) thanks Funda¸caõ para a Ciência e a Tecnologia, and Coimbra Chemistry Centre, Portugal, for support the project UI0313/QUI/2013, co-funded by FEDER/COMPETE 2020-EU.

Supporting Information Available: The final fitting parameters for Eqs. (7), (26), (18), (24) and (25) are gathered in tables S1-6 of supporting information.

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NCO And

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NCO. J. Mol. Spectrosc. 1988, 130, 419–423.

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Figure 10: TOC Graphic

31


Accurate Explicit-Correlation-MRCI-Based DMBE Potential-Energy

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