Potential Energy Surfaces for the First Two Lowest-Lying Electronic

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Potential Energy Surfaces for the First Two Lowest-Lying Electronic States of the LiH_2^+ System, and Dynamics of the H^+ + LiH # H_2^+ + Li Xiaohu He, Shuangjiang Lv, Tasawar Hayat, and Ke-Li Han J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.6b02007 • Publication Date (Web): 29 Mar 2016 Downloaded from http://pubs.acs.org on March 30, 2016

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The Journal of Physical Chemistry

1

Potential Energy Surfaces for the First Two

2

Lowest-Lying Electronic States of the System,

3

and Dynamics of the + ⇌ +

4

Reactions

5

Xiaohu He, †,‡ Shuangjiang Lv*‡, Tasawar Hayat§, , Keli Han‡

⊥

6 †

7

School of Chemical Engineering, Dalian University of Technology, Dalian 116023, China

8 9

‡

State Key Laboratory of Molecular Reaction Dynamics, Dalian Institute of Chemical Physics, Chinese Academy of Science, Dalian 116023, China

10 11 12

§

Department of Mathematics, Quaid-I-Azam University, Islamabad 44000, Pakistan. ⊥

Nonlinear Analysis and Applied Mathematics (NAAM) Research Group,

13

Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah

14

21589, Saudi Arabia.

15 16

AUTHOR INFORMATION

17

Corresponding author

18

*E-mail: [email protected] (Shuangjiang Lv)

19

(Phone: 86-411-84379736, Fax: 86-411-84675584)

1 / 48

ACS Paragon Plus Environment


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Abstract: Two new potential energy surfaces are established for the ground and

2

first excited electronic states of the LiH system, which are important for the

3

astrophysics-related H + LiH and H + LiH reactions. The ab initio energy

4

points are calculated using the complete active space self-consistent field and

5

multi-reference configuration interaction method with aug-cc-pVQZ basis set. At each

6

state, more than 40000 energy points are calculated. The spectroscopic constants of

7

the diatoms and the topographical characters of the new surfaces are examined in

8

detail, showing good agreement with the available literature results. The reaction

9

probabilities, integral and differential cross sections and rate constants for the H +

10

LiH ⇌ H + Li reactions are obtained by performing quantum dynamics

11

calculations, and compared with the previous literature results. The reaction

12

mechanisms are discussed in detail. It is shown that the new surfaces can be

13

recommended for the dynamics study of the H + LiH and H + LiH reactions

14

and other researches including LiH based ro-vibrational spectra and cluster

15

dynamics.

16

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1. Introduction

2

The chemical reactions H + LiH and H + LiH are of considerable

3

importance in the understanding of the primordial lithium chemistry.1-5 In the standard

4

Big Bang model, a gas consisting of H, He, Li and some of their isotopes formed the

5

first stellar objects, implying that the chemistry of the early universe is very simple.4,6

6

The depletion and formation reactions of the LiH and LiH species are proposed to

7

be relevant in the interstellar chemistry and galactic lithium production:1,4,5 the LiH

8

and LiH species are proposed to have contribution to the cosmic background

9

radiation (CBR),1,4-8 because of the high dipole moment of them.6-8 Besides, it is

10

normally considered that LiH and LiH are formed via radiative association,6-9 and

11

LiH can be depleted by interacting with atomic hydrogen or proton,10-14 or undergo

12

elastic or inelastic collisions with atomic hydrogen or helium.15-18 Due to the above

13

considerations, the LiH species are supposed to contribute to the formation of the

14

first condensed structure of the early universe after the recombination era.1,5 Moreover,

15

it is found that LiH at relatively low redshift is mainly ionized.2 Therefore, the

16

reactions involving LiH should be included in a reasonable early universe lithium

17

chemistry model.2,19 In addition to the above astrophysical importance of the LiH

18

system, LiH is also typical because of some physical and chemical concerns. LiH

19

is the simplest ionic chemical system involving a metal atom, and it has only four

20

electrons although the electronic structure is rather complicated.20

21 22

The most probable reactions involved in the LiH system can be these adiabatic processes19,21-25 3 / 48



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1

H + LiH → H + Li

(R1),

2

H + LiH → H + Li

(R2).

3

The reactions R1 and R2 correspond to the ground and first excited electronic

4

states of the LiH system, respectively. For the ground state, it is proposed that the

5

primary pathway is19,22,24 H + LiH → H + H + Li

6

(R3).

7

This collision induced dissociation (CID) process may dominate over the

8

reaction process R1 because of the weak bond of LiH . For the first excited state, the

9

collisions of the reactants H and LiH can be ro-vibrational excitation-deexcitation

10 11 12 13 14 15

processes:19,23,25 H + LiH (v, j) → LiH (v’, j’) + H

(R4).

For the ground state, the H-exchange reaction or non-reactive inelastic collision between H and LiH are possible: H + LiH (v, j) →LiH (v’, j’) + H

(R5).

The above reactions and the reverse of the reaction R2

16

Li + H → LiH + H

17

are supposed to be building blocks in establishing the kinetic details of the early

18

universe.1,4,5,19,24,25 Obviously, the primordial lithium chemistry demands complete

19

and exact dynamic information of the above reactions. To obtain such knowledge, one

20

should perform rigorous quantum dynamics calculations for those reactions. To carry

21

out these studies, accurate full-dimensional potential energy surfaces (PESs)

22

corresponding to different electronic states of the LiH system are prerequisites.14 4 / 48


(R6)

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1

Bodo et al. established a preliminary chemical reaction network of the LiH

2

system by a set of PESs for different electronic states at several specific

3

configurations.21-24 They examined the energetics of the LiH system and proposed

4

several possible chemical reaction channels producing LiH, LiH , H and H .23

5

They pointed out that the destruction of LiH should be efficient comparing the

6

depletion of LiH by proton collision, and the charge transfer process between LiH

7

and LiH is an unlikely pathway since the energy gap between the two lowest-lying

8

states of the LiH system is wide.21,23,24 By means of thoroughly investigating the

9

three lowest-lying PESs of the LiH system for the collinear geometries,24 Bodo et al.

10

found a possible conical intersection region between the first two excited states. The

11

ground-state reaction R1 is expected to be direct, while the reaction R2 for the first

12

excited state should be complex due to the potential wells along the reaction path.24

13

Bodo et al. also performed time-dependent quantum dynamics calculation to

14

investigate the collinear H + LiH collisions, and they found that the CID pathway

15

should dominate over the H + Li pathway.22 Martinazzo et al. further concluded

16

that the conical intersection between the first two excited states is of no direct

17

relevance for the dynamics of low-energy reactions which occur in the first excited

18

state of the LiH system (e.g., R2 and R6).19

19

Martinazzo et al. constructed the first full dimensional PESs of the LiH system

20

for the ground and first excited electronic states.19,25 The PESs were firstly

21

constructed by fitting more than 11000 energy points computed by a multi-reference

22

valence bond (MRVB) approach over the grid of the relevant coordinates (two 5 / 48



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1

internuclear distances and the angle between them).19,22,23,25 Consequently, the PESs

2

were extended with 600 energy points computed at the multi-reference configuration

3

interaction (MRCI) level.19 The PESs of Martinazzo et al.19,25 have been applied to a

4

great number of quantum and classical reactive dynamics studies.10,11,20,26-42 Pino et al.

5

calculated the thermal rate constants for the reactions R1, R2, R3 and R6 using the

6

Quasi-classical Trajectory (QCT) method.26 Bovino et al. performed a series of

7

quantum dynamics calculations and obtained the rate constants for the reactions R1

8

and R2 to understand the kinetics of the LiH system for the two lowest-lying

9

electronic states.10,29,32,41 Roy et al. performed quantum dynamics calculations to

10

study the ground-state reaction R1,33,34 and they also paid special attention on the

11

ultracold H + LiH collision dynamics.34 To obtain the rate constants of the

12

first-excited-state reactions R2 and R6, Bulut et al., Gogtas et al., Akpinar et al., Da

13

Cunha et al. and Aslan et al. also performed a great deal of quantum and classical

14

dynamics calculations.11,20,35-40,42 The stereodynamics of the reactions R1 and R2 are

15

investigated by Li et al.28,30,31 and Duan et al.27 in detail.

16

In this work, two new PESs for the ground and first excited electronic states of

17

the LiH system based on accurate ab initio data are presented. To improve the

18

accuracy of the new PESs, a wider coordinate range and high level quantum

19

chemistry calculation method and basis set are used in the calculations of ab initio

20

energy points. The dynamics calculations for the reactions R2 and R6 on the new

21

first-excited-state PES are carried out by using two recently developed graphics

22

processing

unit

(GPU)

accelerated

time-dependent 6 / 48


wave

packet

(TDWP)

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1

programs.14,43-45 The dynamics results are presented and compared with available

2

literature results. The rest of this paper is organized as follows. Section 2 introduces

3

the ab initio calculations and construction of the PESs. Section 3 describes the

4

spectroscopic constants of the diatoms and topographical features of the PESs. The

5

quantum dynamics calculations and results of the H + LiH ⇌ H + Li reactions

6

are described in Section 4. Section 5 gathered the main conclusions of this work.

7

Finally, we provide the additional useful data in the supporting information.

8

2. Ab initio calculation and construction of the potential energy

9

surfaces

10

The ab initio calculations in this work are carried out using the MOLPRO

11

package.46 The augmented correlation consistent polarization valence quadruple zeta

12

(aug-cc-pVQZ) atomic basis set47-49 is employed for the Li and H atoms. The

13

calculations of the energy points start from a Hartree-Fock (HF) initial guess, and then

14

this guess is provided to the subsequent complete active space self-consistent field

15

(CASSCF)50,51 calculations. Then the ensuing internally contracted MRCI52-54

16

calculations are performed based on the wave functions optimized by the CASSCF

17

procedure. Four electrons are included in the eleven active orbitals (9 a’ and 2 a”) in

18

the CASSCF and MRCI calculations. In the present calculations, the core orbital for

19

the Li atom is not frozen, since the core-correlation effects involving the Li atom are

20

significant. The ground-state energy points are treated adiabatically, because the

21

energy gap between the ground and the first excited states of the LiH system is

22

large in the full configuration space.19,23-25 Even the employed basis set is large, the 7 / 48



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1

standard basis set superposition error (BSSE)55,56 is taken into account in the

2

calculations of the ground-state energy points because the bond of the complex is

3

weak. In the calculations of the first-excited-state energy points, the CASSCF and

4

MRCI calculations are performed by using a three-state averaged method, and the

5

multi-reference Davidson correction57 (+Q) is included to compensate for the effect of

6

higher order correlation. We stress here that the ab initio calculations of the two

7

electronic states are separately performed, therefore the BSSE and Davidson

8

correction are included in the ground and first excited states, respectively.

9

For the two electronic states investigated in the present work, the ab initio energy

10

grid points are generated from the bond coordinates, which are defined by the two

11

bond distances ( and ) and the angle between them (∠LiHH, the included

12

angle of and ). The ∠LiHH angle varies from 0° to 180° with steps of

13

15°. For each ∠LiHH, the ab initio energy points are computed on a × grid.

14

To take into account the long-range contributions to the potentials, the range of the

15

bond length should be large and the grid should be dense. The and

16

distances vary in the intervals of 1.3 – 35.0 Å and 0.6 – 35.0 Å, respectively. 58 and

17

54 grids are made in the and coordinates, respectively. The grids of

18

and are made denser around the vicinity of the equilibrium distances of the

19

diatoms. Note that the grids in the and coordinates are wider and denser

20

than that of the previous works.19,25 For each electronic state, a total of 40716 ab initio

21

energy points are calculated to construct the PES. As a result, a three-dimensional

22

cubic spline interpolation approach58 is employed to yield the PESs for the ground 8 / 48


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1

and first excited electronic states of the LiH system. To ensure the calculated

2

potential energies at large internuclear distances vary correctly and regularly, the

3

energies are modified by multiplying a damping function which has the following

4

form

5

, " , = 1 − ∏1(2 1 − [1 − tanh&'( )( − (,*+, -.]0

(1).

6

( , (,*+, and '( are the internuclear distance, reference geometry and strength

7

parameter, respectively, and 3 = 1, 2 and 3 represent LiH′, LiH" and HH diatoms,

8

respectively. This procedure is done via a thorough examination and comparison of

9

the adjusted energies and the ab initio energies. The root mean square (RMS)

10

deviations resulting from this procedure are less than 0.01 kcal/mole for the two states.

11

The FORTRAN routines and the parameters of the damping function are available

12

upon request to one of the authors.

13

3 Features of the potential energy surfaces

14

In Table 1, the spectroscopic constants of LiH , H , LiH and H obtained

15

from the new PESs are compared with other experimental and theoretical

16

results.19,59-68 The LiH and H species are responsible to the ground state of the

17

LiH system, and the other two species (LiH and H ) are responsible to the first

18

excited state of the LiH system. Overall, it can be found that the present results are

19

in good agreement with these literature results. The newly obtained equilibrium

20

internuclear distances and vibrational constants reproduce the available experimental

21

and theoretical results well. The present dissociation energies of H and H are

22

close to the corresponding literature results. The present dissociation energy of LiH 9 / 48



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1

is slightly higher than the corresponding literature values, and this difference can be

2

attributed to the effect of the non-adiabatic crossing between the first two excited

3

states of the LiH system. The studies of Bodo et al.21,24 and Martinazzo et al.19,25

4

proposed that there is some possible non-adiabatic crossing between the first two

5

excited states of the LiH system near the H + LiH asymptote. In this case, the

6

Mulliken charge distribution21 indicates that the positive charge is not well localized

7

on the isolated H atom, but can be transferred from one H atom to another. Therefore

8

the potential energy corresponding to the H + LiH asymptote should slightly

9

deviate from that of a single LiH molecule (which means that the LiH molecule is

10

not interacting with any other particles). In Table 1, The dissociation energy result of

11

LiH referred from Ref. 64 (fitted from a series of measured spectra values)

12

corresponds to a single LiH molecule. The present calculation confirmed this

13

possibility of the non-adiabatic crossing between the first two excited states of the

14

LiH system, and found that this non-adiabatic crossing has negative effect on the

15

dissociation energy of the LiH moiety. Especially, the agreement between the present

16

dissociation energy of LiH and the literature results is good. Since it is proposed

17

that the positive electrical charge is well localized on the Li atom,19,21 the LiH

18

moiety in the LiH system for the ground state manifests as a single LiH ion.

19

In this work, 0 eV corresponds to the Li − H − H asymptote (i.e., the three-body

20

break-up asymptote) in the discussion about the topographical features of the PESs.

21

Figure 1 illustrates the typical cuts of the new ground-state PES at four ∠LiHH

22

angles (45°, 90°, 135° and 180°). The variation of the contours is quite smooth in the 10 / 48


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1

whole configuration space. In each panel, two valleys are distinguishable: the deeper

2

one on the left corresponds to the H + Li asymptote and the valley at the bottom

3

corresponds to the H + LiH asymptote. In both of the valleys, there is no energy

4

barriers. The potential energy -0.1 eV is highlighted by a dashed line in the contours

5

to indicate the location of the H + LiH asymptote. Observing the figure, one can

6

find that the Li − H − H asymptote is only about 0.1 eV above the minimum of the

7

valley at the bottom, indicating that the CID process is a possible pathway of the

8

collision between LiH and H. The potential energy -4.8 eV is also highlighted by a

9

dashed line. As shown in Figure 1, this dashed line indicates a potential well on the

10

deeper valley of the Figures 1(a), 1(b) and 1(c). In Figure 1(d), no potential energy is

11

below -4.8 eV, therefore the well is shallower than that of the other three

12

configurations. A closer observation reveals that the well is deeper at the angle of 90°

13

than that of the other three configurations. The direct optimization study of Searles

14

and von Nagy-Felsobuki69 and the ground-state PES of Martinazzo et al.19 also

15

supports a shallow potential well lying under the H + Li asymptote.

16

Some typical cuts of the new first-excited-state PES are shown in Figure 2. The

17

variation of the contours is also found to be smooth in the whole configuration space.

18

The left valley corresponds to the H + Li asymptote, and the bottom valley

19

corresponds to the H + LiH asymptote. The PES supports two potential wells, one

20

in the left valley and the other one in the bottom valley, and they are connected by a

21

saddle point. To highlight the locations of the wells, the potential energy of -3.0 eV is

22

portrayed as a dashed line in each panel. At small ∠LiHH angles (e.g., ∠LiHH = 45° 11 / 48



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1

in Figure 2(a)), the wells are shallow, and with the increasing of the angle, the wells

2

become deeper. The minima of the wells and the stationary point are all in H − H −

3

Li collinear geometry. Therefore, the reaction path emerged in Figure 2(d) is

4

essentially the absolute minimum energy path (MEP) of the reaction R2 (or R6). This

5

double-well structure is supported by the studies of Kraemer and Špirko70 and

6

Martinazzo et al.,19 in which the wells and the saddle point are also found to be in

7

H − H − Li collinear geometry.

8

The MEPs in different configurations (at a fixed ∠LiHH angle or obtained by

9

relaxing the ∠LiHH angle) are shown in Figure 3. In Figure 3(a), the absolute MEP

10

on the new ground-state PES is compared with the MEP on the PES of Martinazzo et

11

al.19 The two MEPs are found to be very similar with each other. It is shown that the

12

reaction R1 is highly exothermic and barrier-less, and a shallow potential well appears

13

in the exit channel of R1. The exothermicity of R1 is ~4.596 eV. The well is in C 7

14

geometry, and the depth of the well is 0.266 eV relative to the the H + Li

15

asymptote. The minimum structure of the well (also the global minimum of the PES)

16

indicates the ∠HLiH angle to be 21.35°, the distance to be 0.751 Å and the

17

distance to be 2.0271 Å. Table S1 in the supporting information provides a detailed

18

comparison between the present minimum structure of the potential well and the

19

results reproduced from the studies of Searles and von Nagy-Felsobuki69 and

20

Martinazzo et al.19 The present minimum structure comes closer to the result provided

21

by the direct optimization calculation of Searles and von Nagy-Felsobuki.69 Figure

22

3(b) shows the MEPs at two ∠LiHH angles and the absolute MEP of the present 12 / 48


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1

ground-state PES. For the ∠LiHH angles of 45° and 180° the depths of the wells are

2

0.178 eV and 0.066 eV, respectively. The absolute MEP indicate the ∠LiHH to be

3

79.32°, which is between the angles of the other two MEPs. The MEPs on the new

4

first-excited-state PES and the PES of Martinazzo et al.19 are compared in Figure 3(c).

5

Along the MEP, two wells appear and they are connected by a saddle point. All these

6

critical configurations are in H − H − Li collinear geometry. The two MEPs are very

7

similar with each other, except that the potential wells on the PES of Martinazzo et

8

al.19 is more localized. The well on the left of the new PES is deeper than the

9

corresponding well on the previous PES, and the well on the right of the new PES is

10

shallower, although the difference is very small. Considering a H − H − Li collinear

11

system, the distance between the two hydrogen atoms is defined as and the

12

distance between the middle hydrogen atom and the lithium atom is . On the

13

present first-excited-state PES, the minimum of the well in the entrance channel of the

14

reaction R2 indicates the distance to be 2.517 Å, the distance to be 1.615

15

Å, and the energy to be 1.292 eV below the H + LiH asymptote. The saddle point

16

is 1.104 eV above the minimum of the well in the entrance channel of R2 with =

17

1.614 Å and = 2.558 Å. The minimum of the well in the exit channel of R2 is

18

0.547 eV below the H + Li asymptote with = 1.064 Å and = 3.494 Å.

19

The geometry structure of these critical configurations is found to be well consistent

20

with the calculations of Kraemer and Špirko70 and Martinazzo et al.19 (see Table S2 in

21

the supporting information). In particular, the present result is closer to the recent

22

result of Kraemer and Špirko70 which aims at the spectroscopic properties of the 13 / 48



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1

LiH system for the first excited state. Figure 4(d) shows the MEPs at two ∠LiHH

2

angles and the absolute MEP of the first-excited-state PES. As mentioned before, the

3

absolute MEP on the first-excited-state PES is actually the MEP at the ∠LiHH angle

4

of 180°. As the ∠LiHH angle increases, the depths of the wells and the height of the

5

corresponding saddle point decrease. We note that all major topographical features

6

have been included in the above discussion.

7

To present the topographical features of the new PESs in a different way, Figure

8

S1 in the supporting information shows the energy plots of the two new PESs for an

9

atom moving around a diatom with fixed bond length (the equilibrium internuclear

10

distance of the diatom).

11

4. Dynamics on the new first-excited-state PES

12

To assess the quality of this newly established PES, the dynamics of the H +

13

LiH 8 = 0, 9 = 0 → H + Li (R2) and Li + H 8 = 0, 9 = 0 → LiH + H

14

(R6) reactions are investigated using the GPU accelerated TDWP programs.14,43-45 8

15

and 9 denote the initial vibrational and rotational quantum numbers of the reactant

16

molecule, respectively. The calculations of the reactions R2 and R6 are carried out

17

using the product coordinate-based (PCB)43 and reactant coordinate-based (RCB)44

18

programs, respectively. According to the study of Gómez-Carrasco et al.71, the mass

19

combination of a reaction makes that either PCB program or RCB program is more

20

convenient in the dynamics calculation. Detailed description of the TDWP method

21

can be found in the previous literatures (see Refs. 43, 44 and 45, and the references

22

therein). 14 / 48


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To ensure the convergence of the dynamics results, extensive test calculations

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have been performed. The optimized parameters are thus determined and listed in

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Table 2. The parameters are firstly applied to the calculation for total angular

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momentum : = 0, and then extended to the calculations of : > 0. The included

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collision energies and total angular momenta for the above two reactions are also

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shown in Table 2. To obtain accurate dynamics information, the usage of

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approximations should be restrained. In this work, no total angular momentum is

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omitted in the calculation. The Coriolis Coupling (CC) effect is treated accurately.

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Strict quantum dynamics calculation demands including all of the helicity quantum

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states,72,73 but it would be too time consuming. Normally, it is not necessary to include

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all of the helicity states to obtain converged dynamics result.14,72,73 On the premise of

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ensuring the convergence of the dynamics result, the helicity basis used has been

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truncated according to the relation < ≤ min :,

Potential Energy Surfaces for the First Two Lowest-Lying Electronic

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