Variational calculations of HBN energy levels in the X 2 Π and A 2 Σ + statesMałgorzata Biczysko, Riccardo Tarroni, and Stuart Carter Citation: The Journal of Chemical Physics 119, 4197 (2003); doi: 10.1063/1.1594174 View online: http://dx.doi.org/10.1063/1.1594174 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/119/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Global ab initio potential energy surfaces for both the ground ( X ̃ 1 A ′ ) and excited ( A ̃ 1 A ′ ′ ) electronic statesof HNO and vibrational states of the Renner-Teller A ̃ 1 A ′ ′ – X ̃ 1 A ′ system J. Chem. Phys. 135, 104304 (2011); 10.1063/1.3632994 The electronic spectrum of the fluoroborane free radical. I. Theoretical calculation of the vibronic energy levels ofthe ground and first excited electronic states J. Chem. Phys. 130, 164309 (2009); 10.1063/1.3122008 Energy levels of HCN + and DCN + in the vibronically coupled X 2 Π and A 2 Σ + states J. Chem. Phys. 115, 11200 (2001); 10.1063/1.1414347 Vibrational spectroscopy of phosphaethyne (HCP). I. Potential energy surface, variational calculations, andcomparison with experimental data J. Chem. Phys. 112, 8446 (2000); 10.1063/1.481483 Theoretical study of the potential energy surfaces and dynamics of CaNC/CaCN J. Chem. Phys. 106, 8073 (1997); 10.1063/1.473815

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Variational calculations of HBN energy levels in the X 2P and A 2S¿ statesMałgorzata Biczyskoa) and Riccardo TarroniDipartimento di Chimica Fisica ed Inorganica, Universita` di Bologna, Viale Risorgimento 4,40136 Bologna, Italy

Stuart Carterb)

Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, United Kingdom

~Received 10 April 2003; accepted 2 June 2003!

A theoretical study of the HBN radical in theX 2P andA 2S1 states, taking into account vibroniccoupling effects, is reported. The lowest (12A8,12A9,22A8) potential energy surfaces~PES! of theHBN–BNH system have been studied to identify all stationary points. The HBN minimum wasfound to be 20.0 kcal/mol above BNH, with an isomerization barrier of'11 000 cm21 on the 12A8surface. For the HBN isomer, accurate near-equilibrium three-dimensional diabatic PESs for thegroundX 2P (1 2A8,12A9) and first excitedA 2S1 (2 2A8) electronic states have been calculatedat the multireference configuration interaction level of theory, with extended basis set. A vibroniccoupling between bend and BN stretch, analogous to that found in the isoelectronic C2H and HCN1

radicals, has been found to take place due to the crossing of theX 2P andA 2S1 states at energiesclose to 11 000 cm21. Vibronic energy levels of HBN and DBN have been calculated variationallyusing a previously developed method@Carter et al., Mol. Phys. 98, 1967 ~2000!# suitable forthree-atomic molecules showing three-state vibronic interactions. Energy levels ofS and Psymmetry up to 10 000 cm21 for HBN, and 8800 cm21 for DBN, are reported. It is shown that dueto the high-energy surface crossing, the vibronic interaction becomes non-negligible only for levelsabove 8500 cm21. For all levels, Renner–Teller effects and Fermi resonances are analyzed.© 2003 American Institute of Physics.@DOI: 10.1063/1.1594174#

I. INTRODUCTION

Molecular systems showing nonadiabatic interactions inthe ground state are not very common, but present manyinteresting features and represent a challenging task for boththeoretical and experimental studies. Recently, we developedan approach suitable for theoretical calculation of accuraterovibronic spectra of three-atomic radicals showing three-state interactions.1 The approach has already been success-fully applied to C2H ~Ref. 1! and HCN1.2 In the presentwork we extend these studies to the HBN radical which,being isoelectronic to the previously studied molecules, mayshow similar interactions. At present there are no experimen-tal data for this molecule; therefore, all information must becalculated from first principles.

HBN and its more stable isomer BNH may be obtainedas products from the reaction of boron atoms with ammoniamolecules. BNH was first observed by Knightet al.,3 usingmatrix electron spin resonance spectroscopy, and subse-quently by Andrews and co-workers4 by infrared spectros-copy in argon matrices. A theoretical study of the B1NH3

reaction performed by Wanget al.5 shed some light on thelack of HBN in the postreactive mixture. It was shown thatboth HBN and BNH are exothermic products with respect toisolated boron and ammonia, with relative energies of219.7and 257.3 kcal/mol.5 However, the creation of HBN fromthe intermediate product HBNH2 would require the breakingof two NH bonds; therefore, a very high energy barrier is

expected. Additional formation of HBN from the reaction ofHBNH or BNH with a second boron atom, although pos-sible, is very unlikely to take place. The few other theoreticalstudies of HBN available in the literature do not deal with itselectronic structure or vibrational spectra. Pappovaet al.6

considered HBN as a potential product of H(HCNB)2H andH(HCNB)4H defragmentation in a theoretical study of hy-drogen cyanide borane oligomers and their dehydrogenatedanalogs. Allendorf and Melius7 reported theoretical BAC-MP4 ~bond additivity corrected fourth order Møller–Plessetperturbation theory! thermodynamical data for HBN andBNH, together with other molecules of theB–N–Cl–H sys-tem.

In this work, a theoretical study of HBN vibrational en-ergy levels, including vibronic and spin–orbit coupling, isperformed. In Ref. 1 we have shown that the calculation ofspectra for systems showing nonadiabatic effects requiresseveral specific treatments. First, for molecules showingRenner–Teller and/or Herzberg–Teller interactions, energylevels cannot be obtained by considering only one potentialenergy surface~PES! at a time, because of the failure of theBorn–Oppenheimer approximation. Second, electronic pro-cesses for which the spin–orbit interaction becomes largecannot be treated perturbatively. Moreover, the possibility ofaccidental anharmonic resonances should be taken into ac-count. To meet all these requirements, it is necessary to setup variational calculations which go beyond the Born–Oppenheimer approximation, using the full rovibrationalHamiltonian and considering the coupling between all pos-sible sources of angular momenta~spin, orbital, vibrational,rotational!. Such a procedure, which has proved to predict

a!Electronic mail: bicz@kn.plb!Permanent address: Department of Chemistry, Reading RG6 2AD, UK.

JOURNAL OF CHEMICAL PHYSICS VOLUME 119, NUMBER 8 22 AUGUST 2003

41970021-9606/2003/119(8)/4197/7/$20.00 © 2003 American Institute of Physics

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rovibronic energy levels with high accuracy,1,2 has been setup also for the calculation of vibrational levels of HBN andDBN, with the aim of assisting future experimental work.

In Sec. II we describe the details of theab initio andvariational computations we have performed. In Sec. III theHBN–BNH isomerization is discussed, while Sec. IV is fo-cused on the analysis of HBN near-equilibrium PESs. Thespin–rovibronic energies are reported in Sec. V together withan analysis of vibronic couplings and anharmonic reso-nances.

II. COMPUTATIONAL DETAILS

The lowest 12A8, 1 2A9, and 22A8 electronic surfacesof HBN and BNH have been first investigated to find allstationary points. Geometry optimizations were performed atthe CASSCF level,8,9 using a full valence active space andthe cc-pVQZ basis set.10 The nature of the stationary pointswas confirmed by frequency calculations. For each surface,orbitals have been optimized independently, using state-averaged calculations only when degenerate states were in-volved. Relative energies were determined by single pointenergy calculations at the internally contracted multirefer-ence configuration interaction~ICMRCI! level of theory.11,12

In ICMRCI calculations, all 9 valence electrons were corre-lated with a full valence complete active space~CAS! asreference and the cc-pVQZ10 basis set; in addition, theDavidson size-consistency correction13,14was applied. Allabinitio calculations have been performed using theMOLPRO15

package.Next, for the HBN isomer, three PESs, i.e., 12A8,

1 2A9 (X 2P), and 22A8 (A 2S1) have been mapped for en-ergies up to 11 000 cm21 above theX 2P minimum, neglect-ing in this way the isomerization path to BNH~see Sec. III!.Four hundred seventy-sevenab initio points have been cal-culated withRBN andRBH bond lengths ranging from 1.8 to3.3 bohr and 1.6 to 3.3 bohr, respectively, with angles in therange 180<u<120 deg.

The procedure to calculate accurate diabaticab initioPESs suitable for subsequent variational computations hasalready been tested and applied successfully to C2H ~Ref. 1!

and HCN1.2 Adiabatic energies have been obtained at theICMRCI level of theory with full-valence CAS as referenceand with the Davidson size-consistency correction. Molecu-lar orbitals were obtained from full-valence state averagedCASSCF computations, followed by orbital diabatization.16

For linear geometries only,C2v symmetry properties wereexploited and all 13 electrons were correlated using thecc-pcVQZ17 basis. For a second more extended set of points,including both linear and bent geometries ofCs symmetry,only the 9 valence electrons were correlated and the cc-pVQZ basis set10 was used.

The two adiabatic energies ofA8 symmetry were thendiabatized using the method of block diagonalization of theelectronic Hamiltonian18,19 to get the diabaticV11 (1 2A8)andV33 (2 2A8) energies and theV13 coupling. This proce-dure is based on the construction, for each nonlinear geom-etry, of a 232 transformation matrix from the two leading CIcoefficients issued by the ICMRCI treatment. In this ap-proach it is assumed that, for fixed bond lengths, valenceorbitals maintain for all bent geometries the same physicalcharacter of the linear ones. This condition is achieved bydiabatizing CASSCF orbitals prior to the MRCI, i.e., maxi-mizing their overlap with orbitals obtained at linear geom-etries. This is possible thanks to the invariance of theCASSCF wave function with respect to any orbital rotationwithin the active space.15

In the last step the resulting diabatic energiesV11, V33,as well as the coupling coefficientV13 and theV22 energiesof A9 symmetry have been fitted to an analytic form bymeans of theSURFIT program.20 Three sets of points (V11,V22, andV33) have been fitted to a polynomial expansion inthe form

Vaa~RBN ,RBH ,u!5(i jk

Ci jk~RBN2RBNeq,a! i~RBH2RBH

eq,a! j

3~u2ueq,a!k k even, a51,2,3. ~1!

TheCi j 0 andCi jk ~for kÞ0) coefficients have been obtainedseparately and then merged using the procedure described inRef. 1. The equilibrium geometries andCi j 0 coefficients are

TABLE I. Computed CASSCF/cc-pVQZ geometries and CASSCF/cc-pVQZ//ICMRCI~1Q!/cc-pVQZ energiesfor all stationary points of the 1A8, 2 A8, and 1A9 surfaces of the HBN–BNH system.

CASSCF/cc-pVQZCASSCF/cc-pVQZ//

ICMRCI~1Q!/cc-pVQZ

RBN /bohr RBH /bohr RNH /bohr /HBN/deg Energy/hartreeDE/hartree DE/cm21

1 A8BNH (X 2S1) 2.3534 1.8871 0.0 280.051 121 0.0 0BNH–HBN saddle 2.4698 2.3584 85.0 279.968 593 0.082 528 18 113HBN (X 2P) 2.5030 2.2439 180.0 280.019 211 0.031 910 7 0032 A8BNH (A 2P) 2.6198 1.9136 0.0 279.968 556 0.082 565 18 121BNH-HBN saddle 2.5031 2.5290 55.6 279.865 386 0.185 735 40 764HBN (A 2S1) 2.3693 2.2396 180.0 279.989 925 0.061 196 13 4311 A9BNH (A 2P) 2.6198 1.9136 0.0 279.968 556 0.082 565 18 121BNH-HBN saddle 2.6087 2.4981 66.3 279.922 622 0.128 499 28 202HBN (X 2P) 2.5030 2.2439 180.0 280.019 211 0.031 910 7 003

4198 J. Chem. Phys., Vol. 119, No. 8, 22 August 2003 Biczysko, Tarroni, and Carter

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based on the fitting of theC2v energies alone, while theCs

set, including linear and bent geometries, has been used toget theCi jk (kÞ0) coefficients.

The coupling coefficientsV13 have been fitted to themodified polynomial form

V13~RBN ,RBH ,u!

5sin~u2u ref!• b12tanh~RBN2RBNref !c

• b12tanh~RBH2RBHref !c

3(i jk

Ci jk~RBN2RBNref ! i~RBH2RBH

ref ! j~u2u ref!k, ~2!

usingRBNref 52.1 bohr,RBH

ref 52.1 bohr, andu5180 deg as ref-erence geometry.

The four fitted surfaces have then been used to calculatespin–rovibronic levels by means of the variational programRVIB3.1 The details of theRVIB3 computational procedures,along with the complete form of spin–rovibronic Carter–Handy Hamiltonian,21 are reported in Ref. 1. As in previousapplications1,2 the spin–orbit coupling effect has been intro-duced by a geometry-independent effective spin–orbit con-stantASO. An ab initio value of 244.0 cm21 has been ob-tained at the equilibrium geometry of theX 2P state, usingan ICMRCI wave function22 for the cc-pVQZ basis set withall 9 valence electrons correlated. Calculations of rovibroniclevels up toJ53/2 have been performed for HBN and DBN.The rovibrational basis set has been set up from 44 one-dimensional harmonic oscillators for stretching modes~con-tracted to 52 two-dimensional ones! and 46 primitive associ-ated Legendre functions with no contractions for bendingmode. This basis ensures the convergence of all levels below5000 cm21 to within 0.1 cm21 and up to 10 000 cm21 towithin 0.5 cm21.

III. HBN–BNH ISOMERIZATION

The optimized CASSCF/cc-pVQZ geometries of the sta-tionary points on the lowest HBN-BNH PESs ofA8 andA9symmetry are listed in Table I together with their correspond-ing ICMRCI/cc-pVQZ energies. A schematic representationof the isomerization paths for the states considered here isalso presented in Fig. 1. For HBN, equilibrium geometriesand theX 2P→A 2S1 transition energy have been also de-termined, at a higher level of theory, from ICMRCI/cc-

pcVQZ surfaces~see Sec. IV!. These values can be consid-ered as a best estimate to be compared with CASSCF/cc-pVQZ//ICMRCI/cc-pVQZ results. Indeed, bond lengthsagree to within 0.04 bohr while the transition energy is only66 cm21 higher; hence, we can assume an uncertainty of6100 cm21 for all relative energies listed in Table I.

Literature data to be compared with are quite limited. Allarise from studies at lower levels of theory.3,4 It is interestingto note that while the HF/6-311G~d,p!3 andCASSCF/cc-pVDZ4 geometries agree well~to within 0.02Å! with our results, the relative energies differ considerably.Indeed, at the CIS~D!/DZP level of theory,3 the X 2S1

→A 2P transition of BNH is overestimated by'5000 cm21,while the CASSCF/cc-pVDZ difference of ground-state en-

FIG. 1. Relative energies of stationary points of the 1A8, 2 A8, and 1A9surfaces of the HBN–BNH system.

FIG. 2. ~Color! Contour plots of the PESs at linear geometries for theX 2Pand A 2S1 states of HBN. Energy increments of 1000 cm21 starting fromthe minimum of theX 2P state. The crossing seam is shown as a bold line.

TABLE II. Quartic force field of theX 2P of HBN in internal coordinates.r[RBN , R[RBH , a[sinu. The numbers in parentheses refer to theA9~lowest! surface. All values in aJÅ2n, n being the number of stretchingcoordinates entering into definition of the force constants.

Theoretical

f rr 9.7457f RR 4.3693f aa 0.3604 ~0.2732!f rR 20.0707f rrr 256.8714f RRR 221.8136f rrR 0.0105f rRR 0.1649f raa 0.0791 ~20.3597!f Raa 20.1138 ~20.1019!f rrrr 267.3898

f RRRR 102.4418f aaaa 0.9951 ~1.0968!f rrrR 20.6428f rRRR 21.0164f rrRR 20.8909f rr aa 20.3762 ~20.5222!f RRaa 20.3205 ~20.3566!f Rraa 0.3454 ~0.3487!

4199J. Chem. Phys., Vol. 119, No. 8, 22 August 2003 HBN energy levels

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ergies of HBN and BNH is slightly overestimated by'700cm21.4

Figure 1 shows that the lowest saddle point connects theA8 component of theX 2P state of HBN to theX 2S1 state

of BNH, while the highest isomerization path ofA8 symme-try connects the excitedA 2S1 state of HBN with one com-ponent of theA 2P state of BNH. The lowest barrier toisomerization lies about 18 000 cm21 above the ground stateof BNH, corresponding to'11 000 cm21 with respect to the

FIG. 3. ~Color! One-dimensional cuts of the adiabatic~black, green! and diabatic~violet 1 2A8, red 22A8) PES along BN stretching coordinate, for differentvalues ofu. RBH fixed at 2.1 bohr. The component ofA9 symmetry of the2P state is shown as a blue line.

TABLE III. Calculated energies of HBN, referred to the average of theX(000)P3/2 , X(000)P1/2 , for selected levels ofS andP symmetry up to10 000 cm21. Assignment are based on plots and expansion coefficients ofvibrational part of wave functions~see the text!. All values in cm21.

n DnSO Assignment n DnSO Assignment

0.0 242.5 X ~0,0,0!P 6295.6 X (2,1,0)kS1

624.6 X (0,1,0)mS2 6368.7 A (0,0,0)S1

830.4 X (0,1,0)kS1 6535.7 X (1,1,2)mS2

1309.2 24.0 X ~0,2,0!mP 6847.4 X (1,1,2)kS1a

1550.2 215.5 X ~0,2,0!kPb 6876.1 X (0,1,4)mS2

1630.5 221.2 X ~0,0,1!Pb 7009.2 A ~0,1,0!P1957.2 X (0,3,0)mS2 7090.3 232.0 X ~2,0,1!Pb

2213.8 X (0,1,1)mS2 7379.0 X (0,1,4)kS1

2306.4 X (0,3,0)kS1a 7485.4 231.2 X ~1,0,3!Pc

2468.5 X (0,1,1)kS1a 7632.3 A (0,2,0)S1

2788.4 242.5 X ~1,0,0!P 7797.5 223.2 X ~0,0,5!Pc

3149.0 230.2 X ~0,0,2!Pc 8076.6 X (1,1,3)mS2

3394.5 X (1,1,0)mS2 8122.2 242.4 X ~3,0,0!P3603.7 X (1,1,0)kS1 8139.1 A (0,0,1)S1

3785.9 X (0,1,2)mS2 8263.9 A ~0,3,0!P4062.1 23.9 X ~1,2,0!mP 8287.1 X (1,1,3)kS1a

4104.8 X (0,1,2)kS1a 8586.0 222.6 X ~2,0,2!Pb

4316.7 29.9 X ~1,2,0!kPb 8683.1 X (3,1,0)mS2

4399.7 226.8 X ~1,0,1!Pb 8772.9 A ~0,1,1!P4743.2 224.3 X ~0,0,3!Pc 8905.5 X (3,1,0)kS1

4974.2 X (1,1,1)mS2 8959.6 220.3 X ~1,0,4!Pc

5227.6 X (1,1,1)kS1 9175.7 A (1,0,0)S1

5339.1 X (0,1,3)mS2 9242.1 X ~0,0,6!Pc

5495.8 242.5 X ~2,0,0!P 9393.9 A (0,2,1)S1

5741.6 X (0,1,3)kS1a 9700.9 235.7 X ~3,0,1!Pb

5908.5 227.3 X ~1,0,2!P 9800.2 A ~1,1,0!Pe

6081.0 X (2,1,0)mS2 9896.6 A (0,0,2)S1d

6218.0 X ~0,0,4!Pc 9901.4 X (0,1,5)mS1d

aWeak Fermi resonance betweenX(n1,1,1)kS1 andX(n1,3,0)kS1 states.bVery strong Fermi resonance betweenX(n1,0,1)P and X(n1,2,0)kPstates.

cMember of (n212n3)5even Fermi polyad.dStrong vibronic interaction betweenA(0,0,2)S1 andX(0,1,5)mS1 states.eWeak vibronic interaction betweenA(1,1,0)P andX(2,6,0)kP states.

TABLE IV. Calculated energies of DBN, referred to the average of theX(000)P3/2 , X(000)P1/2 , for selected levels ofS andP symmetry up to8800 cm21. Assignment based on plots and expansion coefficients of vibra-tional part of wave functions~see the text!. All values in cm21.

n DnSO Assignment n DnSO Assignment

0.0 242.8 X ~0,0,0!P 5814.6 X (1,1,2)kS1

503.5 X (0,1,0)mS2 5915.4 227.4 X ~0,0,4!Pa

672.7 X (0,1,0)kS1 6373.4 X (0,1,4)mS2

1055.7 23.8 X ~0,2,0!mP 6387.2 A (0,0,0)S1

1279.1 5.6 X ~0,2,0!kP 6441.2 242.2 X ~3,0,0!P1489.8 241.8 X ~0,0,1!P 6570.2 235.6 X ~1,0,3!P1579.4 X (0,3,0)mS2 6652.3 X (0,1,4)kS1

1881.8 X (0,3,0)kS1 6901.3 X (3,1,0)mS2

1990.0 X (0,1,1)mS2 6908.5 24.3 A ~0,1,0!P2184.0 242.8 X ~1,0,0!P 7043.1 X (1,1,3)mS2

2180.4 X (0,1,1)kS1 7078.8 X (3,1,0)kS1

2672.9 X (1,1,0)mS2 7201.2 X ~2,0,2!P2846.4 X (1,1,0)kS1 7251.7 221.9 X ~0,0,5!Pa

2970.6 239.7 X ~0,0,2!P 7287.3 X (1,1,3)kS1

3212.1 23.7 X ~1,2,0!mP 7413.0 A (0,2,0)S1

3441.6 5.6 X ~1,2,0!kP 7811.1 X (0,1,5)mS2

3464.7 X (0,1,2)mS2 7868.0 239.6 X ~3,0,1!P3654.6 241.8 X ~1,0,1!P 7887.7 212.6 X ~1,0,4!Pa

3680.0 X (0,1,2)kS1 7927.6 24.3 A ~0,3,0!P4140.4 X (1,1,1)mS2 8002.3 A (0,0,1)S1

4330.9 242.7 X ~2,0,0!P 8144.4 X (0,1,5)kS1

4334.7 X (1,1,1)kS1 8411.9 A (0,4,0)S1b

4444.4 235.4 X ~0,0,3!P 8467.5 X (1,1,4)mS2

4804.2 X (2,1,0)mS2 8518.1 237.5 X ~4,0,0!P4930.5 X (0,1,3)mS2 8541.0 A ~0,1,1!Pc

4983.1 X (2,1,0)kS1 8655.5 233.1 X ~2,0,3!P5116.0 239.7 X ~1,0,2!P 8641.0 A (1,0,0)S1

5172.6 X (0,1,3)kS1 8705.3 228.6 X ~0,0,6!Pa

5595.8 X (1,1,2)mS2 8755.1 X (1,1,4)kS1

5780.8 242.3 X ~2,0,1!P

aMember of (n212n3)5evenFermi polyad.bStrong vibronic interaction betweenA(0,4,0)S1 andX(0,2,9)S1 states.cStrong vibronic interaction betweenA(0,1,1)P andX(0,4,4)P states.

4200 J. Chem. Phys., Vol. 119, No. 8, 22 August 2003 Biczysko, Tarroni, and Carter

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HBN ground state. The other two transition states have beenfound at '28 000 cm21 and at '40 000 cm21 above theX 2S1 state of BNH. For BNH, the first excited electronicstate lies more than 18 000 cm21 above theX 2S1 groundstate; therefore, for this isomer vibronic interactions are ex-pected to be negligible. On the other hand the first excitedstate of HBN,A 2S1, is only '6400 cm21 higher in energywith respect to theX 2P state, and vibronic coupling mayplay a relevant role. Since our variational approach can dealonly with a single minimum PES, we had to limit the studyof HBN spectroscopy to energies up to 11 000 cm21 abovethe ground state, that is, tunneling to BNH was not allowed.

IV. THE X 2P AND A 2S¿ SURFACES

The near-minimum diabaticab initio surfaces of HBNhave been fitted to analytical form as described in Sec. II.The resulting expansion coefficients are deposited as supple-mentary material,23 while the internal quartic force field forthe X 2P, recovered from surface parameters neglecting thevibronic coupling is presented in Table II. The equilibriumstructures of theX 2P and A 2S1 states have been deter-mined from the fit of linear geometries, while the differenceof the correspondingC000 coefficients has been taken as thebest estimate of theX 2P→A 2S1 pure electronic transitionenergy Te . These values are, for theX 2P state, RBN

eq

52.4769,RBHeq 52.2055 bohr, and, for theA 2S1 state,RBN

eq

52.3550,RBHeq 52.2031 bohr;Te50.028 984 hartree~6361.3

cm21!.In Fig. 2 we show contour plots of the

X 2P (1 2A8,12A9) andA 2S1 (2 2A8) states, together withtheir crossing seam. It can be seen that the crossing takesplace at rather high energies, compared to isoelectronic C2Hand HCN1; thus, the effects of vibronic coupling on thelowest levels are expected to be small.

The one-dimensional cuts of adiabatic and diabatic sur-faces along the BN stretching coordinate are plotted in Fig.3. The plots correspond to bending angles fixed at 180, 170,and 150 deg, respectively, with the BH distance held fixed to2.2 bohr. It can be seen that even if theA 2S1 state lies only'6400 cm21 above theX 2P state, the potential surfaces docross at much higher energies. This reduces considerably thevibronic interactions between the two states, which, forslightly bent geometries (u5170 deg), is limited to a smallregion around the crossing seam. Much larger interactions,extending over the full PESs, are found for strongly bentgeometries (u5150 deg).

V. THE ROVIBRONIC ENERGY LEVELSOF HBN AND DBN

The rovibronic energy levels have been calculated varia-tionally according to the scheme described in Sec. II. Se-lected states ofS andP symmetry are presented in Tables IIIand IV for HBN and DBN, respectively. For boron, the mostabundant isotope has been considered. Only levels withJ5P, whereJ is the total angular momentum andP is thesum of orbital and spin angular momenta, are reported. ForP levels, which are split by spin–orbit interaction, the aver-age energy ofJ53/2 andJ51/2 levels is reported, whileDnSO refers to theP3/22P1/2 energy difference. Levelslisted in Tables III and IV have been selected on the basis oftheir chance of experimental detection; in particular, stateswith extended bending excitation and complex combinationbands are omitted. The assignments are based on the plotsand the expansion coefficient of the pure vibrational part ofthe wave functions. For resonant levels only the dominantcomponent is quoted together with the type of interaction.Additional information about all levels ofS andP symmetryup to 10 000 cm21 for HBN and 8800 cm21 for DBN aredeposited as supplementary material.23

For this radical there is no experimental information tocompare. Hence, the expected accuracy of theoretical rovi-brational levels can be judged only from previous studiesusing the same approach. In the case of C2H ~Ref. 1! andHCN1,2 the pure electronic transition energiesTe wereslightly adjusted~within 650 cm21! to obtain a better agree-ment with experimental data. For HBN such adjustment isnot possible; hence, while the levels of theX 2P state areexpected to be accurate within 10–15 cm21, as in previousstudies, for the levels belonging to theA 2S1 state the shiftsmay be as large as 50–100 cm21.In the low-energy region all levels belong to theX 2Pground electronic state, which, being doubly degenerate, isaffected by Renner–Teller and spin–orbit interactions. Ne-

FIG. 4. Vibrational wave function of rovibronic states ofP symmetry andJ51/2, belonging to then52 Fermi polyads of HBN. The plots representcuts of the pure vibrational part of the total spin–rovibronic wave functionalong two internal coordinates, namely the bending~x axis! and the B–Nstretching~y axis!, while RBH is kept at its equilibrium value. Positive andnegative lobes are plotted with continuous and dashed lines, respectively.The three electronic components (151 A8, 251 A9, 352 A8) are in adja-cent panels with the contribution from each electronic state to the total wavefunction shown in the upper right corner of each panel.

4201J. Chem. Phys., Vol. 119, No. 8, 22 August 2003 HBN energy levels

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glecting the vibronic coupling with theA state, the Renner–Teller splitting parameter« and the harmonic bending fre-quencyv2 are defined as

«5~f2212

2f2222

!/~f2212

1f2222

!, ~3!

v251

&A~f22

121f22

22!, ~4!

wheref226 are bending force constants in normal coordinates

~for HBN: f221 5672.01,f22

2 5771.85 cm21) and1/2 referto the upper (A8) and lower (A9) surfaces, respectively. Us-ing the values quoted above, we have«50.138 andv2

5721.7 cm21, which causes a splitting of them andk levelsof more than 200 cm21, substantially larger than the spin–orbit splitting.

Other interesting features of the spectra are the anhar-monic resonances found for both HBN and DBN. Variationalcalculations automatically take into account all possible in-teractions and provide information about vibrational wavefunction shape and coefficients, making possible an analysisof resonances. At a qualitative level, the degree of mixingand the strength of the anharmonic interaction can be judgedfrom the plots of the vibrational wave function.

In the case of HBN, a classical Fermi resonance involv-ing levels with n212n35n has been found. Especiallystrong mixing has been observed for levels (n1,0,1) and(n1,2,0), and such interaction is shown in Fig. 4 for themembers ofn212n352 polyad. It is evident that Fermiresonance is dominated by the interaction between~0,0,1!P–~0,2,0!kP states, and that the wave functions are stronglymixed. For othern212n35n polyads, for even values ofn,strong mixing of all states formally belonging tok ~upper!surface within the polyad has been found. On the other hand,for odd values ofn, states ofk origin are only slightly per-turbed and unambiguous assignments are always possible.For DBN, low energy levels remain essentially unperturbedand anharmonic interactions within Fermi polyads are foundonly for (n212n3)>6, in the dense state region.

The ~0,0,0! of theA 2S1 electronic state has been foundat 6368.7 cm21 for HBN and 6387.2 cm21 for DBN. As

discussed in the previous section, theX 2P andA 2S1 elec-tronic states cross more than 11 000 cm21 above the mini-mum of theX 2P state. However, some relevant vibronicinteraction can be predicted even at lower energies. The mostnoticeable case of such mixing takes place for theA(0,0,2)S1 andX(0,1,5)mS1 states of HBN as depicted inFig. 5~a!; in Fig. 5~b! a much weaker interaction between theA(1,1,0)P and X(2,6,0)kP states is portrayed. For HBNthere is no other state perturbed below 10 000 cm21. In thecase of DBN there are two pairs of states showing strongvibronic interactions below 8800 cm21; these areA(0,1,1)PandX(0,4,4)kP, along withA(0,4,0)S1 andX(0,9,2)S1.

VI. CONCLUSIONS

A theoretical study of spin–rovibronic levels for HBNand DBN radicals in theX 2P and A 2S1 has been per-formed, taking into account the vibronic coupling betweenthe two states. Energy levels ofS and P symmetries up to10 000 cm21 for HBN and 8800 cm21 for DBN have beencalculated variationally based on fullyab initio diabaticPESs. All levels belonging toX 2P state withn2.0 are splitinto m andk components due to a Renner–Teller interactionwhich is dominant over the spin–orbit effect. For HBN,Fermi resonance withinn212n35n polyads has been foundwith mixing of all states formally belonging to thek ~upper!surface. No such interaction has been observed for DBN. Forboth HBN and DBN only few vibronically perturbed stateshave been found; this makes HBN quite different with re-spect to the isoelectronic radicals C2H and HCN1, for whichvibronic perturbations dominate the whole spectrum.

ACKNOWLEDGMENTS

This work was supported by the MURST of Italy and theEEC under the RTN Program 7~Contract No. HPRN-CT-1999-00005!. M.B. and S.C. would like to acknowledge CI-NECA, Casalecchio di Reno, Bologna, Italy for the award ofa MINOS grant~EC/TMR program!. All authors are gratefulto P. J. Knowles and H. -J. Werner for providing them withaccess to theMOLPRO code and to Professor P. Palmieri for

FIG. 5. Vibrational wave function of rovibronic states of~a! S and ~b! P symmetry, showing vibronic interactions for HBN. The meaning of the symbolsexplained in the caption of Fig. 4.

4202 J. Chem. Phys., Vol. 119, No. 8, 22 August 2003 Biczysko, Tarroni, and Carter

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suggestions and stimulating discussions. M.B. would like toacknowledge the Wroclaw Supercomputer Center for provid-ing computer time, where some of the calculations were per-formed. S.C. would like to acknowledge support from theU.S. Office of Naval Research.

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Mol. Phys.98, 1823~2000!.23See EPAPS Document No. E-JCPSA6-119-313332 forX 2P (1 2A8,

1 2A9) andA 2S1 (2 2A8) PES expansion coefficient and complete list ofS andP energy levels for HBN and DBN. A direct link to this documentmay be found in the online article’s HTML reference section. The docu-ment may also be reached via the EPAPS homepage~http://www.aip.org/pubservs/epaps.html! or from ftp.aip.org in the directory /epaps/. See theEPAPS homepage for more information.

4203J. Chem. Phys., Vol. 119, No. 8, 22 August 2003 HBN energy levels

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