Chemical Physics Letters 389 (2004) 101–107

www.elsevier.com/locate/cplett

A coupled treatment of 1Rþ and 3P states of AgH molecule

Yan Li a, Heinz-Peter Libermann a, Robert J. Buenker a, Luk�a�s Pichl b,*

a Bergische Universitaet Wuppertal, Fachbereich C, Theoretische Chemie, Gaussstr. 20, Wuppertal D-42119, Germanyb Foundation of Computer Science Laboratory, University of Aizu, Ikki, Aizuwakamatsu, Fukushima 965-8580, Japan

Received 29 March 2003; in final form 26 February 2004

Published online:

Abstract

Ab initio configuration interaction (CI) calculations have been carried out for the potential curves of the lowest five 1Rþ and

three 3P electronic states of the AgH molecule. Nonadiabatic couplings among the 1Rþ states and spin–orbit interaction matrix

elements between the 1Rþ and 3P states have been evaluated. The resulting adiabatic potential curves and couplings are transformed

to a diabatic representation by employing a unitary transform which eliminates all d=dR coupling terms. Energy positions and

predissociation rates for vibrational levels associated with the above electronic states are determined by employing a complex scaling

approach based on both the adiabatic potential curves and their diabatic counterparts and the associated nonadiabatic couplings. It

was found that the differences between these two sets of results for vibrational spacings and predissociation rates are marginal. The

calculated spectroscopic constants for the X1Rþ, A1Rþ and a3P states are in good agreement with measured results, and the cal-

culated vibrational spacings for the A1Rþ state are also in reasonably good agreement with experiment. The reasons behind the

relatively large discrepancies in the predicted and measured Te values for the c3P and B1Rþ states are discussed. Predissociation

linewidths are predicted for the vibrational levels of these electronic states. The decay of the A1Rþ and B1Rþ states is caused by

nonadiabatic effects, whereas that of the a3P and c3P states is induced by the spin–orbit interaction.

� 2004 Elsevier B.V. All rights reserved.

1. Introduction

The avoided crossing of the A1Rþ and the X1Rþ

states of the AgH molecule results in unusual vibrational

characteristics for the A1Rþ state. Experimental and

theoretical studies were carried out in the last tens yearsto understand the details of that perturbation. Bengts-

son and Olsson [1] first determined spectroscopic pa-

rameters for the A and X states by measuring the

emission spectrum of the A1Rþ–X1Rþ transition. They

found that the vibrational spacings of the A state were

not able to be described in terms of the usual Dan Helm

formula. Learner [2] first suggested that the anomalous

behavior of the A1R state was caused by an avoidedcrossing between two 1Rþ states arising from the 5p and

4d states of the Ag atom. This explanation was not

supported by a subsequent experimental investigation

by Ringstr€om and Aslund [3], however. In related the-

oretical studies, Hess and Chandra [4] carried out a spin-

* Corresponding author. Fax: +81-242-37-2734.

E-mail address: lukas@u-aizu.ac.jp (L. Pichl).

0009-2614/$ - see front matter � 2004 Elsevier B.V. All rights reserved.

doi:10.1016/j.cplett.2004.03.067

free relativistic calculation on the A1Rþ potential curve

near the equilibrium internuclear distance of the AgH

molecule, and Witek et al. [5,6] carried out ab initio

calculations for the 1Rþ and 3P states of the AgH

molecule by applying their multi-state multi-reference

perturbation theory. They found that the avoidingcrossings among the 1Rþ states are caused by an

admixture of covalent and ionic character in the inter-

nuclear distance range of 4–6 �A [6]. Vibrational calcu-

lations based on their uncoupled adiabatic potential

curves were also carried out by these authors. In the past

few years our group has carried out accurate ab initio

studies of the low-lying electronic states of a large series

of diatomic molecules containing heavy atoms by em-ploying relativistic effective core potentials (RECPs)

[7,8]. It was decided to employ the same approach to

deal with the nonadiabatic coupling effects for the 1Rþ

states of the AgH molecule. In the present study ab

initio calculations have been carried out for the adia-

batic potential curves of five 1Rþ and three 3P electronic

states of this system. In addition, nonadiabatic cou-

plings among the 1Rþ states and spin–orbit interaction

102 Y. Li et al. / Chemical Physics Letters 389 (2004) 101–107

matrix elements between the 1Rþ and 3P states have also

been evaluated. A unitary transform of the five adiabatic1Rþ states was also carried out to obtain the analogous

results in a diabatic representation, which eliminates all

d=dR coupling matrix elements in order to obtain slowlyvarying quantities. Finally, the complex rotation meth-

od was then employed for both the adiabatic potential

curves and their diabatic counterparts, including the

corresponding nonadiabatic couplings in each case, in

order to determine the energies and predissociation

linewidths and non-radiative lifetime for the vibrational

levels associated with the electronic states mentioned

above.

Table 1

Ag excitation energiesa

Configuration T ¼ 0 Full CI Experiment [22]

4d105s1 0 0 0

4d105p1 29 337 30 329 30 165

4d95s2 32 702 31 659 32 031

4d106s1 42 653 44 030 42 556

4d106p1 46 470 48 005 48 432

aAll in cm�1.

2. Theoretical formulation and computational details

In a close coupling treatment of electronic states, the

total wave functions describing the electronic and nu-

clear motion are expanded in terms of appropriate basis

functions [9],

wð~r;RÞ ¼Xn

vðaÞn ðRÞ/nð~r;RÞ: ð1Þ

In the adiabatic electronic basis Helð~r;RÞ/nð~r;RÞ ¼�nðRÞ/nð~r;RÞ, the Schr}odinger equation transforms as�� �h2

2ld2

dR2

�þ E

�I þ DðRÞ

� �h2

2l� 2AðRÞ d

dR

�þ BðRÞ

��~vðaÞðRÞ ¼ 0: ð2Þ

Here l is the reduced mass, arrows denote vectors, and

all matrices appear underlined. In particular, Iij ¼ dij,Dij ¼ �idij, Aij ¼ h/ijd=dRj/ji, and Bij ¼ h/ijd2=dR2j/ji.Following Heil [10–12], the d=dR operator can be re-

moved by a suitable rotation in the v space,

~vðaÞðRÞ ¼ CðRÞ~vðdÞðRÞ; withd

dRC þ AC ¼ 0: ð3Þ

In addition, the condition for completeness B ¼ A2 þdA=dR can be used to simplify Eq. (2), resulting in�� �h2

2ld2

dR2þ V ðRÞ � EI

�~vðdÞðRÞ ¼ 0; ð4Þ

with

V ðRÞ � C�1ðRÞDðRÞCðRÞ:The diabatic representation in Eq. (3) is fixed by re-

quiring ~vðaÞð1Þ ¼~vðdÞð1Þ, and found by numerical in-tegration,

CðRÞ ¼ I þZ 1

RAðR0ÞCðR0ÞdR0: ð5Þ

In the present work both Eqs. (2) and (4) are employed

to determine the vibrational energies and predissocia-

tion rates of the various electronic states by using

Gauss–Hermite quadrature in conjunction with the

complex scaling method.

The ab initio multi-reference single- and double-ex-

citation configuration interaction (MRD-CI) method is

employed to obtain the adiabatic potential curves andnonadiabatic couplings. The underlying theory and

computer programs have been described elsewhere

[13–20]. The MRD-CI calculations for the AgH 1Rþ and3P electronic states are carried out with the aid of

RECPs [8], whereby only the 4s, 4p, 4d and 5s electrons

of the Ag atom need to be considered explicitly. The

spin–orbit interaction matrix elements between 1Rþ and3P states are also computed by employing an effectivespin–orbit potential [8]. An atomic [5s5p4d] gaussian

basis set from [8] is used for the Ag atom, augmented by

sets of [3f2g] polarization functions and [2s2p3d] Ryd-

berg functions. The exponents of the latter functions

have been determined by energy minimization. A

(6s3p2d) primitive basis contracted to [4s3p2d] is used for

the H atom [21]. The excitation energies for 5s, 5p, 6s and

6p orbitals, both the experimental [22] and calculatedvalues, are shown in Table 1. It can be seen that for the

first two excited states, which prove to be heavily in-

volved in forming the A1Rþ, B1Rþ and a3P states of the

AgH molecule, the energy positions calculated with the

above method are in good agreement with the experi-

mental results. The somewhat larger discrepancies in the

results for the Ag 6s and 6p orbitals are not surprising,

since the higher Rydberg states tend to correlate easily.Test calculations have shown that the spin–orbit in-

teractions between 1Rþ and 3D states are negligible.

Therefore, in this study only 1Rþ and 3P states of 0þ Xsymmetry have been considered. The molecular orbitals

resulting from self-consistent field (SCF) calculations for

the lowest 3Rþ state of AgH are taken as the one-elec-

tron basis for the subsequent CI treatment. A threshold

of 10�7 Eh is used to select configurations, which leadsto secular equations of the order of about 600,000 on the

average for the 7 roots considered totally in 1A1 sym-

metry and internuclear distances from 2.0 to 10.0 a.u.

The effects of unselected configurations on the total

energy are estimated by using a perturbative procedure

[14,15]. The Hamiltonian matrix in the configuration

space is constructed using the table CI algorithm [16–18]

and a direct CI procedure is used to search for the de-sired eigenvalues and eigenvectors [20].

Fig. 1. The calculated potential curves for low-lying 0þ states of AgH:

full lines, the adiabatic potential curves of 1Rþ states; dashed lines, the

diabatic counterparts for 1Rþ states; dash–dot lines, the potential

curves of 3P states.

Y. Li et al. / Chemical Physics Letters 389 (2004) 101–107 103

The radial couplings AijðRÞ among the 1Rþ states are

computed from the highly correlated wavefunctions by

employing the finite difference numerical method with

the step DR ¼ 2� 10�4 a.u. [23,24]. Since numerical in-

stability was encountered in calculating the corre-sponding second-derivative radial couplings BijðRÞ, it

was decided simply to omit these terms for the close-

coupling equations in the adiabatic basis. The errors

introduced by omitting the BijðRÞ terms will be discussed

subsequently.

The calculated adiabatic potential curves and non-

adiabatic couplings for the five 1Rþ states are then

transformed to a diabatic representation employing theprocedure discussed above. The calculated spin–orbit

matrix elements between 1Rþ and 3P states are also

subjected to the same unitary transform. The vibrational

levels and predissociation rate of the 0þ states of the

AgH molecule are determined by using the complex

rotation method [25] for both the adiabatic potential

curves and their diabatic counterparts, including the

respective nonadiabatic couplings in each case. Thecomplex Hamiltonian matrix elements are computed by

employing Gaussian–Hermite quadrature in a basis of

250 complex Hermite polynomials for each electronic

state. Series of scaling angles in the range of 1–5� havebeen considered in order to search for the stabilized

complex eigenvalues corresponding to resonances.

3. Results and discussion

3.1. Potential curves and nonadiabatic couplings

The calculated adiabatic potential curves for the five1Rþ states and their counterparts resulting from the di-

abatization procedure are shown in Fig. 1 along with the

potential curves calculated for the three 3P states. In-spection of the wave functions of the 1Rþ states shows

that their electronic configurations vary dramatically

with changing internuclear distance of the AgH mole-

cule. At long internuclear distances both the X and A

states occupy the H 1s orbital with a single electron. The

X state correlates with the Ag atomic ground state, with

one electron occupying the 5s orbital. The A state dis-

sociates to the lowest Ag excited state, with 5p occupiedinstead of 5s. Thus both asymptotes consist of neutral

atoms. The character of the X and A states changes

significantly with decreasing R, however, so that the X

state becomes closed shell in character near its equilib-

rium distance, and A becomes a valence state with two

open shells. Table 2 shows the change in electronic

configurations with varying R.The third 1Rþ state, the B state, has an even more

complex variation in its composition as the internuclear

distance is varied. At relatively long values (larger than

10.0 a.u.), its wave function is predominantly Rydberg

in character, with a small contribution from the Ag d9s2

H 1s1 configuration. At short distances (less than 3.5

a.u.) its composition is purely Rydberg (Ag 5p and 6s

orbitals). In the intermediate range of distance, however,

its wave function is a mixture of the Ag d9s2 H 1s1 andAg d9s1 H 1s2 configurations.

As can be seen from Fig. 1, the minima of the A1Rþ

and B1Rþ states lie higher in energy than the asymptote

of the X1Rþ state. Hence, non-radiative decay of the A

state and predissociation of the B state can occur by

virtue of nonadiabatic effects related to the aforemen-

tioned dramatic changes in configurations of the above

electronic states with internuclear separation. Moreover,the 3P states are also subject to decay because of spin–

orbit interactions with these 1Rþ states. Nonadiabatic

couplings among the 1Rþ states, crucial for predicting

the non-radiative decay and predissociation, are shown

in Fig. 2. Also the diabatic potentials and couplings are

shown in Figs. 1 and 3, respectively.

The diabatic transform does not have a very large

effect on the X ground state’s potential curve (seeFig. 1). This can be explained by analysis of the calcu-

lated nonadiabatic coupling term A12ðRÞ between the

first two 1Rþ states (Figs. 2 and 3) which show that the

A12ðRÞ curve does not posses a maximum. Rather, its

absolute value increases nearly monotonically with de-

creasing internuclear distance similar to Demkov type

coupling [26] (note that the sign of such an off-diagonal

Table 2

R-variations in leading configurations of 1Rþ statesa

R (a.u.) MO X A B C

2.0 1r21d22r2 0.84 0.06

1r21d22r13r1 0.07 0.84

1r21d22r14r1 0.68 0.15

1r21d22r15r1 0.07 0.20

1r21d22r17r1 0.07 0.50

3.0 1r21d22r2 0.75 0.11

1r21d22r13r1 0.14 0.71

1r21d22r14r1 0.60 0.19

1r21d22r15r1 0.12 0.11

1r21d22r17r1 0.07 0.30

1r11d22r23r1 0.14

4.0 1r21d22r2 0.55 0.19

1r21d22r13r1 0.31 0.53

1r21d22r14r1 0.64

1r21d22r15r1 0.17

1r21d23r2 0.10

1r11d22r23r1 0.35

1r11d22r13r2 0.32

5.0 1r21d22r2 0.35 0.32

1r21d22r13r1 0.54 0.34

1r21d22r14r1 0.61

1r21d22r15r1 0.16

1r21d22r18r1 0.13

1r11d22r23r1 0.13

1r11d22r13r2 0.68

6.0 1r21d22r2 0.15 0.43

1r21d22r13r1 0.77 0.14

1r21d22r14r1 0.51

1r21d22r18r1 0.19 0.17

1r21d22r111r1 0.10

1r11d22r13r2 0.82

15.0 1r21d22r2 0.30 0.21

1r21d22r13r1 0.93

1r21d22r14r1 0.34 0.17

1r21d22r15r1 0.05 0.37 0.29

1r21d22r17r1 0.12

1r21d22r18r1 0.47 0.05 0.26

a 1r corresponds asymptotically to Ag 4dz, 1d to Ag 4dx2�y2 , 2r to H 1s, 3r to Ag 5s, 4r to Ag 5p and so forth.

104 Y. Li et al. / Chemical Physics Letters 389 (2004) 101–107

matrix element simply depends on the relative phase of

the two electronic wave functions), but is never of very

large magnitude. Since the rotational angle that defines

the diabatic unitary transform, as discussed in the pre-

vious section, is determined almost exclusively by theradial dependence of this coupling term, the end result is

that the diabatic potentials deviate only slightly from the

original adiabatic (CI) curves (Fig. 1), except for the A

state for R < 4 a.u. A numerical check of the d=dRmatrix elements in the diabatic representation from Eq.

(3) confirms that all such couplings vanish with the ac-

curacy of �10�5 a.u., which is controlled by the size of

integration and differentiation steps DR.Since the dominant configurations for the X and A

states change rapidly with R, it is interesting that the

d=dR coupling is not more substantial between them.

The key to understanding this unusual behavior lies in

the fact that the AijðRÞ coupling term is composed of two

clearly defined subterms [27–29]. The first involves de-

rivatives of the CI coefficients (CI term), whereas the

second arises from changes in the molecular orbitalsthemselves (MO term). In the present case it is found

that the CI term reaches a definite maximum near

R ¼ 3:0 a.u. (see the dotted curves in Fig. 2). The reason

that the total A12ðRÞ curve does not show a similar

maximum is that in this case the corresponding MO

term has a protracted minimum at almost exactly the

same internuclear distance, leading to a considerable

cancellation effect.Ultimately, the real reason for the absence of a po-

tential crossing of the diabatic X and A states lies in the

condition used to define the diabatic transform in the

Fig. 2. The evaluated hijd=dRjji terms among 1Rþ states (full lines).

The dashed lines denote subterms of A12 (see text).

Fig. 3. The diabatic counterparts of nonadiabatic couplings among1Rþ states (see text).

Y. Li et al. / Chemical Physics Letters 389 (2004) 101–107 105

present study, namely so that the associated d/dR matrix

element vanishes at each value of the internuclear dis-

tance. In order to eliminate any possibility of an artefact

in the present theoretical treatment, we have carried out

additional calculations employing a different SCF-MO

basis. The result is that the CI and MO terms differ

significantly from those in Fig. 2 but their sum is very

nearly the same at each R value as that obtained with theoriginal set of one-electron functions.

3.2. Vibrational spacings and predissociation linewidths

The vibrational energy levels and predissociation

linewidths are determined by using the complex rotation

method in both the adiabatic and diabatic representa-

tions discussed above. It was found that the resultingtwo sets of vibrational energy levels are nearly identical

to one another (differences are always less than 2 cm�1).

Similarly, for almost all vibrational levels the differences

between the two sets of calculated predissociation line-

widths are marginal, except for the v ¼ 0 level of the

B1Rþ state, for which the omitted BijðRÞ terms probably

play a non-negligible role when the vibrational calcula-

tion is carried out in the adiabatic basis. Since the B1Rþ

state configuration rapidly changes with R as discussed

above, details of the potential function near the mini-

mum energy along with the limited number of the cou-

pled states considered may cause this difference.

The calculated equilibrium internuclear distances

(Re), electronic transition energies (Te), rotational con-stants (Be) and harmonic frequencies (xe and xexe) forthe various AgH electronic states are given in Table 3together with the available data from experiment and

other theoretical studies. As can be seen, the calculated

spectroscopic constants for the X1Rþ, A1Rþ and a3Pstates are in very good agreement with the correspond-

ing experimental results. They also represent an im-

provement over the previous theoretical studies [6].

However, there are larger discrepancies between the

calculated and measured Te values for the c3P and B1Rþ

states. The errors in the predicted Te values for these twostates are roughly 2900 and 6500 cm�1, respectively (see

Table 3). In view of the fact that the corresponding

discrepancies in the present computed Te values for theA1Rþ and a3P states are less than 400 cm�1 (see Table

3), it seems probable that the much larger errors of

thousands of cm�1 found for the theoretical Te values ofthe c3P and B1Rþ states are caused to at least someextent by incorrect experimental assignments. Consis-

tent with this assessment is the fact that our predicted Tevalue for the c3P state of 44 602 cm�1 is very close to the

measured value of 44 510 cm�1 reported for the B1Rþ

state (see Table 3). Based on the present calculations it

therefore appears quite likely that this experimental as-

signment should be changed to the c3P state.

The calculated vibrational spacings and predissocia-tion linewidths for the various AgH electronic states are

given in Table 4 for comparison with available experi-

mental results. For clarity only the computed results for

Table 4

Calculated vibrational spacings DGv and predissociation linewidths Cv for low-lying 0þ states of AgHa

v A1Rþ a3P c3P B1Rþ

DGv Exptl.b Cvc DGv Cv DGv Cv DGv Cv

0 8.4� 10�4 4.0� 10�5 7.0� 10�4 5.0� 10�1

1 1444 1490 1.2� 10�3 1385 1.2� 10�2 1351 1.2� 10�3 933 1.6

2 1221 1316 7.6� 10�4 1125 4.0� 10�2 1233 5.4� 10�3 831 3.4

3 931 1109 1.1� 10�3 930 3.6� 10�2 1097 1.4� 10�3

4 736 892 1.6� 10�5 766 4.4� 10�2 933 1.0� 10�3

5 654 735 6.0� 10�4 625 4.4� 10�2 745 7.2� 10�3

6 629 669 2.2� 10�4 488 4.0� 10�2

7 623 646 3.8� 10�4 349 6.6� 10�2

aOnly diabatic results are given, see text. All numbers in cm�1.bRef. [30].c Inverse lifetime for A state which undergoes a non-radiative decay.

Table 3

Calculated spectroscopic constants for low-lying 0þ states of AgH and comparison with experimental resultsa

Teb Re Be xe xexe

X1Rþ This work 0 1.614 6.36 1819 80

Ref. [6] 1.564 6.90 2073 53

Exp. [30] 1.618 6.45 1760 34

A1Rþ This work 30 321 1.640 6.17

Ref. [6] 32 208 1.717 5.73 1422 27

Exp. [30] 29 959 1.638 6.27 1664 87

a3P This work 41 564 1.571 6.63 1634 122

Ref. [6] 44 305 1.594 6.64 1620 89

Exp. [30] 41 700 <1.64 >6.3 1450 50

c3P This work 44 602 1.800 5.22

Ref. [6] 48 490 1.845 4.96 1198 86

Exp. [30] 47 472 1.85 >4.95

B1Rþ This work 50 982 2.011 3.75

Ref. [6] 54 236 2.050 3.86 1026 87

Exp. [30] 44 510 1.862 4.87 1819 65

a For A, B1Rþ and c3P states calculation failed to obtain xe and xexe constants because of the irregularities of their potential curves. All numbers

in cm�1, except Re in �A.bThe dissociation energy of the X1Rþ state is calculated as 2.34 eV versus the corresponding experimental value [30] of 2.28 eV.

106 Y. Li et al. / Chemical Physics Letters 389 (2004) 101–107

the diabatic representation are presented there. In gen-

eral, the predicted vibrational spacings for the A1Rþ are

in reasonably good agreement with the corresponding

measured results. The relatively large errors (about 20%)

in the calculations of vibrational spacings for some in-

dividual vibrational levels (v ¼ 3–5) of the A state can be

ascribed to the effect of the dramatic changes in the

composition of the A state with internuclear distance. Itis not unusual to find that a higher level of electron

correlation is required in such cases.

The computed predissociation linewidths for the vi-

brational levels of the various AgH states treated are

generally very small. Also the non-radiative decay life-

time for the A1Rþ state (see Table 4) is very large. Ex-

amination of the calculated results indicates that this is

due to the relatively small nonadiabatic couplings

among the lowest three 1Rþ states of this system (see

Fig. 2), and also the small overlaps computed between

vibrational wave functions associated with the excited

electronic states and the continuum levels of the X1Rþ

state. Consistent with this finding is the fact that there is

no experimental report of predissociation of the excited

states of the AgH molecule.

4. Conclusion

The ab initio MRD-CI calculations for potential

curves of the first five 1Rþ and three 3P electronic states

of AgH have been carried out by employing multi-ref-

Y. Li et al. / Chemical Physics Letters 389 (2004) 101–107 107

erence CI calculations and RECPs. In addition, non-

adiabatic couplings among the 1Rþ states and spin–orbit

interaction matrix elements between the 1Rþ and 3Pstates have been computed with the resulting CI wave

functions. The adiabatic potential curves and couplingelements have been transformed to their diabatic coun-

terparts, and vibrational calculations have been carried

out using the complex rotation method in both the

adiabatic and diabatic representations. It is found that

the differences between the two sets of values for vi-

brational energies and predissociation linewidths are

marginal.

The calculated spectroscopic constants for the elec-tronic states treated are generally in good agreement

with available experimental results, and represent an

improvement over previous theoretical studies. The

relatively large discrepancies found between calculated

and observed Te values for the c3P and B1Rþ states have

been attributed to an incorrect experimental assignment.

The calculated vibrational spacings for the A1Rþ state

are in reasonably good agreement with observed data.Finally, the complex rotation treatment indicates that

the A1Rþ and B1Rþ states are only weakly predissoci-

ated as a result of the nonadiabatic radial coupling. The

a3P and c3P states are also weakly predissociated, in

this case by both radial coupling and the spin–orbit in-

teraction with the various 1Rþ states.

Acknowledgements

One of us (L.P.) acknowledges the partial support

by JSPS Grant-in-Aid for young scientists. The support

of the Deutsche Forschungsgemeinschaft (Grant BU

450/7-3 and BU 450/15-1) is also hereby gratefully

acknowledged.

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