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Radiative Lifetime of the "Forbidden" a1Πg ← X1Σg+ Transition of N2

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1986 Phys. Scr. 33 135

(http://iopscience.iop.org/1402-4896/33/2/007)

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Physica Scripta. Vol. 33,135-140,1986

1 + Radiative Lifetime of the ”Forbidden” a’n, - X Zg Transition of N2

Flemming Dahl and Jens Oddershede

Department of Chemistry, Odense University, DK-5230 Odense M, Denmark

Received July 19,1985; accepted October 2.5,1985

Abstract

Using the second order polarization propagator method we have calcu- lated the band oscillator strengths of the “forbidden” Lyman-Birge- Hopfield band system, alng +X’,X!g+, of N, and the radiative lifetime ~(zJ‘) of the alng state for 0 < v ‘ < 4. We find that ~ ( 0 ) = 58.6 psec and T ( 4 ) = 6 8 . 2 ~ s . About 95 per cent of the intensity are magnetic dipole contributions and only about 5 per cent come from the electric quadru- pole radiation.

1. Introduction

The Lyman [ l ] -Birge-Hopfield [2] band system (dug * XIZi) is one of the most extensively studied transitions of N2. Even though the transition is electric dipole forbidden, it is readily seen in emission as well as in absorption. It attains its intensity from magnetic dipole (Ml) and electric quadrupole (E2) radiations. It has long been known that the magnetic dipole radiation gives the largest contribution to the intensity. However, since the first rough estimate by Wilkinson and Mulliken [3] who used matrix elements for H2 to calculate a ratio of 0.15 for Ml/E2, there have been suggestions ranging from 0.33 [4] to 0.04 [SI for the same ratio. The present investigation favours the low end of this range and the transition is thus almost solely a magnetic dipole transition.

There have been several experimental investigations [5-15] of the radiative lifetime of the a’ll, state. The measurements show a span of 5-lops [8] to 170ps [6]. The most recent vibrationally resolved decay studies suggest that ~(0) is of the order 1OOpsec. This is almost a factor of two larger than the outcome of the present calculation, ~ ( 0 ) = 5 8 . 6 ~ ~ . It is clear from Fig. 1 that the a’ II, state will decay almost solely into the ground state. We can thus disregard branching into other lower lying states in the calculation of the radiative lifetime of a’n,.

2. Method

The radiative lifetime and the band oscillator strengths are computed using the second order polarization propagator method. This is a direct method for determining energy differ- ences and transition matrix elements [16-181. That is, rather than energies E,, E, and wavefunctions \k, , \k, we compute directly AEnm = En - Em and M,, = (*,I M I *m) where M is any one-electron operator. The excitation energies, AE,,, are obtained as poles of the polarization propagator while the residues of the propagator determine the transition matrix elements. If we express M in second quantization as

then the transition matrix elements between the excited state I i ) and the ground state lo) are

(3)

It is thus only M,, that varies with the computed transition matrix elements. The ( i l a~a , lo ) elements which we obtain from the eigenvectors of the principal polarization propagator [ 191 are the same.

There are several approximation schemes for the polariz- ation propagator. The one we shall use here is based on a perturbation expansion of the propagator through first and second order in the fluctuation potential. Through a given order means that all perturbation terms in that and lower orders as well as certain other higher order terms, often summed to infinite orders, are included in the perturbation expansion [20]. The fluctuation potential is defined as the electron-electron

E in cm-’

I 120,000

100,000

a0,ooo

60,000

40,OOO

20.000

(

INTERNUCLEAR DISTANCE (A)

Fzg. 1. Potential energy curves of N, for which Te < 80 0000 cm-’ cal- calculated from the experimental spectroscopic constants given by Huber and Herzberg [ 261 as described in Ref. 25.

pylysica Scripta 33

136 Flemming Dahl and Jens Oddershede

repulsion minus the average ground state Fock potential. The zeroth order approach is thus a calculation where excitation energies are obtained as Hartree-Fock orbital energy differences. The first order polarization propagator methods turns out [20] to be identical to the well-known random-phase approximation (RPA) and time-dependent Hartree-Fock (TDHF) methods. The second order polarization propagator approximation (SOPPA) is almost [21] the same as the equation-of-motion (EOM) method [ 2 2 ] , but has otherwise no direct resemblance to standard methods of computational physics. If we disregard the B ( l ) matrix [ 181 in RPA, we recover the momoexcited CI (MECI) approximation which thus, using the present order concept, lies somewhere between a zeroth and a first order approximation.

The radiative lifetime r ( i ) of an excited I i) state is obtained as

(4 )

where we have used that gi = 2 and lav ' ) and IXv") label vibrational states of the a'@, and XIXi states, respectively. Since the Lyman-Birge-Hopfield band system is a parity forbidden transition, the first term (El) in eq. (5) vanishes.

The vibrational wavefunctions which we need in eq. (14) were computed using the Numerov-Cooley method [24] as described previously [ 2 5 ] . We could use the shape of one of the potential energy curves together with the computed excitation energies as the potential in the two nuclear SchrBdinger equations. However, since the energy difference to the third power enters in eq. (14), a small error in 2m 1 , x v f g will introduce a large error in the lifetime. Even with the best available theor- etical methods it is not possible to match the accuracy with which spectroscopists can measure energy differences for states which are so well characterized as the a'n, and X'X; states of N2. It is therefore standard procedure in lifetime calculations to use experimental values for 5 and the vibrational wavefunctions. We have also followed this line here and the two states are computed using the spectroscopic constants given by Huber and

where the sum extends over all states If, with energy lower than I i ) and where the Einstein coefficients for spontaneous emission -..a

Herzberg [ 2 6 ] . The band oscillator strength (5,251 is related to the Einstein

coefficient for spontaneous emission. More precisely the relation

LUG

64n4 , p f f [l(iliiE1lf)l2 + I ( ~ M M ~ I ~ ) I ~ Aif = -

3 hgi L is

(8)

where the five tensor components of any of the quadrupole operators q j expressed in Cartesian coordinates are

The three terms inside the bracket give the electric dipole, the magnetic dipole, and the electric quadrupole band oscillator strengths. The quadrupole oscillator strength can also be written as

ME2 = e qj i

mCi2E$jA',fv~~j'Au 20e6h2gfv, J ~ P A "

(9) f/%'A',fvtt$'A!o = - ( iv'A'J'1 ME I fv"A"J" ) I 2 q ( f 2 ) = f (x2 - y 2 * i(xy +yx) )

q(+ 1) = 4 {xz + zx * i(yz + zy)) (10) (16)

Note that the convention adopted here for the quadrupole tensor means that q(0 , * 1 , * 2 ) all are normalized to one I231 .

Invoking the Born-Oppenheimer approximation we wr;' the initial and final states as

and the sum in eq. (4) thus becomes a sum over v" and J". If we average over the rotational states J" and set J' = A('II,) = 1 , we obtain the following expression for the vibrational life- time of the a'n, state

where a = e 2 / k is the fine structure constant. Equation (16) is the expression given by Pilling et al. [5] .

To compute 7 from eq. (14) andf from eq. ( 1 5 ) we need to know the transition moments which, using eqs. (12) and ( 1 3 ) , can be written as

(av'lMMIIXvrr) = (v ' lMgl lvr ' ) (17) where the electronic transition moment

(18) M a x M1 - - (a' ngl hiM1 I x ' is of the form given in eq. ( 3 ) and thus can be calculated from the residues of the polarization propagator. A similar relation holds for the electric quadrupole transition moment. The one- electron basis set in which we expand the Fock orbitals consists of Cartesian Gaussians type orbitals (GTO) and the M,, matrix elements of eq. (2) are thus linear combinations of matrix

Physica Scripta 33

Radiative Lifetime of the "Forbidden"a'n, f X'X: llansition of N2 137

Table I. Excitation energies, Ae, and transition moments, M, for the a'n, f X'E; transition of N2 as afinction of the size of the basis set (R = re = 2.0 7434 au).

~~ ~ ~~ ~ ~~

1 [301 ( 9 3 + [3,21 t ( 2 , 2 ) c ~ ~ 36 -108.880144 9.12 1.64 1 1.285 1.285 2 [311 (995) -c [4,31 t (2s2)CMb 44 - 108.89 1442 9.18 1.636

(11,7,2) + [6,3,21 54 -108.896612 9.73 1.622 1.321 1.615 1.331

1.293 5 [341 (12,7,2) + [6,5,21 66 -108.984443 9.79 1.619 6 [351 (12,8,3,1) + [6,4,3,11 92 -108.985039 9.74 1.619 1.326

3 [321 4 1331 (11,7,3)-+ [5,4,21 58 -108.983444 9.82

a Primitive ( ) and contracted GTO's [ 1. Two p and two d functions placed at centersf-mass. Total number of contracted GTO's. For the Xl Z; state (in au). First order propagator (RPA) results. In units of the Bohr magneton B e . ' Numerical value of the transition moment in one direction, l (u ' I Ig~lZylXIZ~) l for M1 and ItulIIg,IxzlX1 Zi ) l for E2 [see also eqs. (24) and (251.1

elements of the form

(GTOr IMIGTOj) (19)

M = r x p or M = r r (20)

with

and with

GTOi = @*(A, a , I, m , n) = NxLyTzI exp (-a& (21) being a GTO centered on A . The quadrupole matrix elements

ME2 ( A Y @ij

= J @i(A ,cy19 l i , mm>xUcr%zg@j(B, a j , lj, mj, nj)dxdydz (22)

where A , B , and C label three centers, can readily be calculated using standard methods [27] . Using relations like

hCCdf(A, a , 1, m, n ) = - i h y C { n @ f ( A , a , l , m , n - 1)

-2a@t(A,a,I,m,n + 1))

+ &zc{m4i(A, a, I , m - 1, n )

- 2a@i(A, a, 1, m + 1, n)) (23) with IC = - %rcxVc we see that the matrix elements of the 1-term in the magnetic dipole operator, eq. (7), are obtained from the same basic integrals as the quadrupole terms in eq. (22). The I, integrals, for instance, is a sum of 4 integrals of the type given in eq. (22) each multiplied with appropriate constants as shown in eq. (23). Similar relations hold I , and I , and the whole (GTOrIIIGTOj) is thus a sum of 12 terms of the form given in eq. (22).

3. Computational aspects

The calculations were performed with the Munich program system [28]. The preliminary step in a propagator calculation consists of a self-consistent Field (SCF) calculation for the ground state. The SCF molecular orbital energies and integrals are then used to construct the propagator matrices, the diag- onalization of which yield the excitation energies and transition moments. Details about the present propagator program,

including timings for the various steps in the procedure, are given elsewhere [29].

Like in more standard ab initio calculations we must choose a oneelectron basis set in which we then expand the molecular orbitals. We have tested six different basis sets that have been used in earlier N2 calculation. The results are summarized in Table I. Based on these results and consideration of compu- tational feasibility we have chosen to use basis set no. 4 in the rest of the calculations.

Another important parameter that determines the compu- tational effort [29] of a polarization propagator calculation is the number of doubly occupied molecular orbitals (7 in the case of N,) and the number of doubly occupied virtual orbitals. In the present implementation of the propagator programs [29] we are using Abelian point group symmetry, i.e. D2h is the most symmetric point group that we can make use of. We thus only need to consider particle-hole excitations which transform as one of the irreducible representations of D2h. For the Lyman-Birge-Hopfield band that means excitations of either B2, or B3, symmetry which will then come out as being degenerate (the two components of the Z; + llg excitations in the DZh point group). In Table I1 we have examined the effect of reducing the size of the excitation space. Clearly, disregarding excitations from the two lowest molecular orbitals (1s on N) do not affect the outcome of the propagator calculations. Further- more, this truncation greatly reduces the computational effort

Table 11. RPA excitation energies, Ae, and transition moments, M, for the a'n, t X'Z; transition o f N2 as a function of the size of the particle-hole excitation spacea.

1-7b 8-58b 43b 9.82 1.614 1.331 3-7 8-58 32 9.82 1.615 1.331 5 -7 8-58 21 10.13 1.604 1.407 3-7 8-56 31 9.82 1.617 1.331 3-7 8-52 29 9.82 1.617 1.331 3-7 8-42 21 9.96 1.647 1.398

R =re = 2.07434 au and using basis set 4 of Table I Full excitation space within the considered basis set. Number of excitations of either B2, or Bsg symmetry in D2h. In units of the Bohr magneton De. Numerical value of the transition moment in one direction, It4'IIgx12,1 XIZj)l for M1 and lKalIIg,lxzlXIZ~)l for E2 [see also eqs. (24)and (2511.

Physica Scripta 33

138 Flemming Dah1 and Jens Oddershede

since it reduces the number of particle-hole excitations by 11. If a similar reduction in the excitation space originating from truncation of the virtual orbital space should be achieved, Table I1 shows that we would need to remove so many virtual orbitals that it would affect the accuracy of the calculation. We have therefore chosen to use the excitation space given in the second row of Table 11, i.e. disregarding only excitations out of the two 1s core orbitals.

The results of the calculations are given in the next section.

4. Results and discussion

The excitation energies, the magnetic dipole (Ml), and the electric quadrupole (E2) transition moments were computed at the three levels of approximation that were discussed in sec. 2: MECI, RPA (first order), and SOPPA (second order). The results are given in Tables I11 and IV and plotted in Fig. 2 . We see that the excitation energies are clearly best in the second order polarization propagator approach and that they in this order are very close to the experimental potential energy difference. Only at the largest internuclear separation do we see the onset of the nonconvergence of the perturbation expansion. However, this point can be omitted from the lifetime calculations without changing the results.

No other calculations exist of electronic transition moments for the Lyman-Birge-Hopfield band, and experimentally it is only possible to obtain transition moments using approximate methods to interpret the primary experimental results, the radiative lifetimes. Thus, only the results of the present calcu- lation is given in Table IV. We see from Fig. 2 that both tran- sition moments vary with the internuclear separation: MMl goes through a maximum while MEZ decreases with increasing R. The three levels of approximation give nearly the same result with RPA and SOPPA being most different.

A note on conventions is probably in order. We use real molecular orbitals n, and rY. Choosing the internuclear axis as the z-direction we thus have two identical non-vanishing M1

Table 111. Excitation energies, Ae (in eV), of the a' llg + A'' transition of N2 as a function of the intemuclear separation, R a .

R A€

au A MECI RPAb SOPPA' Experimentd

1.90000 2.00000 2.03000 2.06700 2.07434 2.19208 2.30546 2.57000 2.80000 3.00000 3.25000

1.005 11.74 11.54 10.72 1.058 10.76 10.54 9.88 1.074 10.48 10.25 9.63 1.095 10.11 9.96 9.32 1.098 10.07 9.82 9.22 1.160 9.01 8.72 8.42 1.220 8.05 7.70 7.69 1.360 6.01 5.48 6.36 1.482 4.47 3.61 5.59 1.588 3.27 1.78 5.09 1.720 1.97 (e) 4.35

10.76 9.89 9.67 9.38 9.31 8.51 7.79 6.43 5.62 5.17 4.84

a Using basis set 4 of Table I. First order polarization propagator results. Second order polarization propagator results. Calculated as the difference between the experimental potential energy curves, which were obtained from the spectroscopic constants given by Huber and Herzberg [ 261 by the method outlined in Ref. 25.

e The RPA solution is singlet instable (imaginary excitation energy) at that distance.

Table IV. Magnetic dipole ( M l ) and electric quadrupole (E2) transition moments (in au) for the a'ns 4 X'Z; trrmsition o f Nz as a function of the intemuclear separation, Ra.

R M"ii8 M i 2

au A MECI R P A ~ SOP PA^ MECI R P A ~ SOP PA^ 1.90000 1.005 1.456 1.554 1.495 1.533 1.444 1.614 2.00000 1.058 1.479 1.591 1.521 1.491 1.376 1.554 2.03000 1.074 1.485 1.602 1.529 1.474 1.377 1.537 2.06700 1.094 1.492 1.615 1.538 1.452 1.334 1.516 2.07434 1.098 1.493 1.615 1.542 1.450 1.331 1.515 2.19208 1.160 1.511 1.656 1.561 1.393 1.267 1.460 2.30546 1.220 1.523 1.694 1.574 1.344 1.212 1.418 2.57000 1.360 1.542 1.814 1.574 1.203 1.095 1.343 2.80000 1.482 1.552 2.032 1.539 1.166 0.989 1.280 3.00000 1.588 1.562 2.696 1.452 1.102 0.819 1.221 3.25000 1.720 1.584 (e) 1.281 1.021 (e) 1.106

a Using basis set 4 of Table I. See footnotes b, c of Table 111.

' Numerical value of the transition moment in one direction, I(a'ngxl lylX'Z$l for M1 and Ita'ngxlxzlXIZ;)I for E2 [see also Eqs. (24) and (25)]. In units of the Bohr magneton Be.

e The RPA solution is singlet instable (imaginary excitation energy) at that distance.

E in cm-'

120,000

100,000

80,000

60,000

40,000

20,000

1.3

1.2

1.1

x '1's \ t j I I I I I I I

INTERNUCLEAR DISTANCE (A)

Fig. 2. The two potential energy curves and the magnetic dipole ( M M ~ in units of the Bohr magneton P e ) and electric quadrupole ME^) transition moments of the Lyman-Birge-Hopfield, a' lTg + X' Xi, band system of N,. The potential energy curves are taken from Fig. 1 and the transition moment are computed in the second order polarization propagator approximation. See footnote c of Table IV for the definitions of M M ~ and &2. Vibrational levels of a ' lTg(w ') and X' Xi (Y ") are indicated.

Physica Scripta 33

Radiative Lifetime of the "Forbidden"a'll, +X'Zi Transition of N2 139

components (a ' I Igy l lx~XIZ~) and ( a ' l l g x ~ l y ~ X ' Z ~ ) both of which are the same as the matrix elements for complex orbitals

of the two tensorial components of 1. If we use the same convention for M l transitions as Whitting et al. [36] suggested for E l transitions, i.e. eq. (24), then the transition moments in Tables I, 11, and IV are what Witting et al. called the electronic transition moments, R e , for a ll+

For the E2 transition the two q(f 1) components in eq. (lo), are non-vanishing for the Lyman-Birge-Hopfield band. Since q(f 1) are normalized tensor components [like l/fl(lx k ily)], one should in analogy with the E l and M1 transitions define the electronic transition moment for an electric quadrulpole tran- sition as

transition.

Table V. Vibrational band oscillator strengths (x IO') of the a'l l , +X'Z; transition o fNtTb.

U" 0 1 2 3 4

0 6.16 17.04 25.80 28.36 25.24 1 21.02 27.57 14.39 1.99 0.83 2 33.06 10.99 0.46 10.82 14.25 3 32.14 0.07 14.80 9.78 0.10 4 21.61 11.28 11.26 0.52 10.81

a Computed in the second order polarization propagator approach using basis set 4 of Table I.

The sum of the M1 and E2 terms in eq. (15). More details are published in the thesis of F. Dahl, which is available from the authors upon request.

R,(E2, ll +E) = (alllgflq(f 1)IX'Zi)

= fl(a'II,IyzIX'ZB+)

= fl(a'n,,IxzIX'Z:) (25)

The last equalities are obtained using the relations between complex and real orbitals together with eq. (10). The quad- rupole matrix elements in Tables I, 11, and IV are thus Re(E2,

Using the experimental potential energy curves and the procedure outlined in sec. 3 [24-261 we have calculated band oscillator strengths and Einstein coefficients. The results are given in Tables V and VI. It is evident that the Lyman-Birge- Hopfield band primarily obtains its intensity from magnetic dipole radiation. The second order results show that only 4 4 % of the intensity originate from the E2 matrix exlements. Wilkinson and Mulliken [3] made the first estimate of this ratio using matrix elements computed for H2 and calculated AM'/ AE2 to be 0.15. Shemansky [lo] found that the ratio varied between 8-10% and the measurements of Pilling et al. [SI gave 4-7%. Both experiments were performed for 0 < v' < 6 and especially the findings of Pilling et al. agree well with the best of our calculations (SOPPA).

In Table VI1 we have compared the calculated, and the measured lifetimes and we see that also for this quantity do we find the best agreement with Pilling et al. [5]. However, with exception of Pilling et al. [ 5 ] and a couple of the first measure- ments, most experiments seem to predict that the lifetime of the a'ng state is about a factor of two larger than what we found in the present calculation. The most recent measurements of Curtis and Erman [15] gives ~ ( 1 ) = 120 f 20psec, i.e. a factor of two larger than the calculated lifetime. However, the actually measured lifetime [15] was 35 k 3ps, but due to thermal escape effects Curtis and Erman argued that the real lifetime should be a factor of -3.4 larger. If, perhaps, the

n + Z)/fi.

Table VI. Einstein coefficients and radiative lifetimes of the a' ll, + X'Z; transition of N;lb.

A,E?IA$?

V I RPA' SOPPAd RPA' SOPPAd RPA' SOPPAd RPA' SOPPAd

0 19 108 16 399 488 662 0.026 0.040 5 1.03 5 8.6 1 1 18 790 16 137 512 701 0.027 0.043 51.81 59.39 2 18 467 15 887 516 748 0.028 0.047 52.68 60.11 3 17 934 15 396 5 23 760 0.029 0.049 54.18 61.90 4 16 126 13 945 514 7 26 0.032 0.052 60.10 68.16

a Using basis set 4 of Table I. More details about individual Einstein coefficients are published in the thesis of F. Dahl, which is available from the authors upon request. First order polarization propagator approach. Second order polarization propagator approach.

Table VII. Comparison of calculated and measured radiztive lifetimes (in p ) of the Lyman-Birge-Hopfield band of N2.

Experiments Calc.

Y ' [6Ia [7ia [sia [io1 [Illa [12Ia [13]' [ 141 a [51 [151 SOPPA

- 59 0 170f30 120f50 31 144*43 80 150 115 * 20 105 * 35 - 1 146 f 44 76* 10 120f20 59 2 148 r 44 9Or 8 60 3 150 * 45 98 r 8 62 4 152 f 46 - 68

a Vibationally non-resolved measurements.

Physica Scripta 33

140 Hemming Dah1 and Jens Oddershede

thermal escaoe corrections were somewhat smaller, this would 9. Ching, B. K., Cook, G. R. and Becker, R. A., J . Quant. Spectrosc. bring their measurements more in agreement with our calcu- lation and the curve-of-growth measurements of Pilling et al. [ 5 ] . Obviously, more calculations and experiments will be needed to establish the magnitude of the lifetime of the a’& state.

5. Summary

Using the first (RPA) and second order (SOPPA) polarization propgator approximation we have computed the radiative life- time of the a’II, state of N?. The state decay via magnetic magnetic dipole (Ml) and electric quadrupole (E2) radiation to the X’X; ground state. We find that E2 is only responsible for about 4-5% of the intensity. The Lyman-Birge-Hopfield band system is thus almost solely a magnetic dipole transition. We calculated the lifetime of a’& to be ~(0) = 58.6psec which is about a factor of two smaller than the most recent measure- ments [15] but only about 25% less than ~(0’) measured by Pilling et al. [5] .

Acknowledgement

We thank Jan Geertsen for computational assistance.

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5 .

6 . 7 .

8.

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pytvsica Scripta 33