ACCURATE CALCULATIONS OF THE DYNAMIC DIPOLE POLARIZABILITY OF N2.
A MULTICONFIGURATIONAL LINEAR RESPONSE STUDY USING RESTRICTED ACTIVE SPACE (RAS) WAVEFUNCTIONS Hans Jsrgen Aa. JENSEN, Poul J0RGENSEN, Trygve HELGAKER
Department of Chemistry, Aarhus University, DK~5000 Aarhus C, Denmark.
and
Jeppe OLSEN
Theoretical Chemistry, Chemical Center, University oflund, P.O. Box 124. S-221 00 Lund, Sweden Received 26 May 1989; in final form 10 August 1989
The dynamic polarizability of N2 is calculated using multiconfigurational linear response (MCLR) with restricted active space CI expansions. This allows inclusion of the static correlation and part of the dynamic correlation. The calculations which include dynamic correlation give results close to the experimental values with only a small fraction of the orbital space occupied.
1. Introduction
The dynamic polarizability is of fundamental im- portance for a description of Rayleigh scattering, Ra- man scattering, collision-induced excitation and many other phenomena. Some progress has been made in developing configuration interaction (CT) and many-body perturbation theory (MBPT) meth- ods for the accurate calculation of dynamic response properties; however, these methods remain rather cumbersome, since the orbital response is included through higher-order excitations. Although single and double excitations from the reference space can de- scribe the correlation accurately, triple excitations must be included in order to describe the combined effect of correlation and orbital relaxation. A more promising approach has been the second-order po- larization propagator approximation (SOPPA) where the linear response function is calculated through second order in the fluctuation potential [ 11.
due to a time-dependent external field is obtained from a projected time-dependent Schriidinger equa- tion. Only the excitations necessary for describing the electron correlation are needed in the CI expansion.
The MCLR equations defining the orbital and CI responses have recently been formulated in a direct fashion [ 4,5 1, and suitable iterative algorithms have been devised [6] and implemented in our general MCSCF program system [5,7]. The program em- ploys a new set of CI codes [ 81 that support CI ex- pansions of the restricted active space (RAS) type.
In RAS the partly occupied orbital space, the active orbital space, is divided into three subspaces: RASI, RADII and RASIII. The allowed configurations are restricted by imposing a lower limit to the number of electrons in RASI, and an upper limit to the al- lowed number of electrons in RASIII. No special re- quirements are put on the orbital occupancy in RASH.
The multiconfigurational linear response (MCLR) Large scale MCLR calculations were recently per- method [2-41 offers a viable alternative. Starting formed on CH+ and compared to full CI(FC1) re- from a fully optimized multiconfigurational self- sults [9]. Various RAS constructions, including consistent field (MCSCF) wavefunction the relax- complete active space [lo] (CAS), were used in ation of the occupied orbitals and the CI coeffkients MCLR calculations with up to 400000 determi-
0 009-2614/89/% 03.50 0 Elsevier Science Publishers B.V. 355
Table I
Contracted Cartesian Gaussians for N (six Cartesian d functions are used)
Function
IS
1s 1s 1s 1s 1s 1s 1s 2P
2P 2P 2P 2P 2P 3d 3d 3d
Exponent Coefficient 5909.44 0.002004
887.451 0.01531 204.749 0.074293
59.8376 0.253364 19.9981 0.600576
7.1927 1.0 2.683 1.0
0.7 1.0
0.2133 1.0 0.07 1.0 0.025 1.0 0.007 1.0 26.786 0.018257
5.91564 0.116407 1.7074 0.390111 0.5314 1.0 0.1654 1.0
0.05 1.0
0.015 1.0 0.005 1.0
0.88 1.0
0.22 1.0
0.07 1.0
nants. These calculations demonstrate that (i ) cor- relation treatments beyond valence CAS are neces- sary to obtain highly accurate results and (ii) suitable RAS constraints can significantly simplify the cal- culations without impairing the accuracy, compared to CAS calculations.
In this communication we employ the MCLR method to calculate the dynamic dipole polarizabil-
ity and polarizability derivatives of Nz. The dynamic polarizability of N, is experimentally known for a wide range of frequencies and the anisotropy deriv- ative has been measured experimentally at 5 14.5 nm [ 111. Several accurate calculations of the static po- larizability have been published as well as several less accurate dynamic polarizabilities [ 12- 15 1. Calcu- lations of the anisotropy derivative predict a larger value than reported experimentally. Our calculations
support this prediction.
2. Calculations
All calculations used a 12s8p3d set of primitive GTOs contracted to 8s6p3d. The basis is given in ta- ble 1. The 12~8~ part is the 9s5p basis of Huzinaga [ 161 augmented with three diffuse s functions and three diffuse p functions. The three innermost s and p functions were contracted according to Dunning [ 171. The exponents of the three six-component d shells are those given by Werner and Meyer [ 181.
No f functions are included in the basis set. As shown by Langhoff et al. [ 121 this introduces errors of less than 0.1 au in the dipole polarizability.
At the internuclear distance rN_N= 2.074 bohr our basis set gives cxll(0)= 15.00 au andcx, (0) =9.79 au at the coupled Hartree-Fock (CHF) level. This compares well with the recent CHF results of Jaszunski et al. [13], a,(O)=I5.03 au and ells (0) = 9.82 au, as well as with other good basis set values [ 121.
A systematic study of the effects of correlation on the static polarizability was carried out at rN_N= 2.068 bohr and the results are given in table 2. The follow-
Table 2
MCLR frequency-independent polarizabilities (atomic units) of N2 using various active orbital spaces
Active space ci Y
30,20.lw1 11.49 5.16
20*20”ll~lu~l~“l~l~ls 10.99 4.06
2~,1~“//12~*2~“1~2el’ 10.77 4.85
la,lo.//[( )~(2~~lslr.)(2a‘2a”2x,2n,l6‘)~]‘O 11.54 4.82 la,la,// [ (20,10”)4( 17Qrr”) (20,2u”17&2nJsg)~]‘o 11.58 4.65
experiment 11.76” 4.45 b)
‘) Ref. [l9]. b)Ref. (201.
ing notation is used for RAS calculations. The num- ber of inactive orbitals in each symmetry class is fol- lowed by a double slash (// ) and a pair of brackets in which three pairs of parentheses indicate the or- bitals contained in RASI, R4SII and RASIII, re- spectively. The superscript on the bracket denotes the number of electrons in the active orbital space.
The superscript following the parentheses of RASI indicates the smallest number of electrons allowed in RASI, while the superscript following the parenthe- ses of R4SIII gives the largest number of electrons allowed in RASIII. In CAS calculations the round parentheses are omitted. The RAS calculation de- noted lo,la,//[(20,10,)4(l~~lrr,)(20,10,1rc,lx,- 1Q2] ID is thus a calculation with an inactive space consisting of one as orbital and one aU orbital. The calculation has ten electrons in the active orbital space. The RASI space contains two 0, orbitals and one q, orbital with a minimal occupancy of four electrons and the RASII space contains one ~cg and one n, orbital. The RASIII space can at most contain two electrons.
The choices of CAS and RAS orbitals were based on an MP2 analysis of the occupancies of the natural orbitals [ 2 11. This provides a simple way of ensur- ing well balanced active spaces for a system such as Nz. The 30,20,1 n,// calculations are CHF calcula- tions. The CHF isotropy differs by 2W from the ex- perimental value, while the CHF anisotropy differs by 16%. In the 2og20U//[ lo,l~~l~~l~,]” calcula- tions the 2p orbitals are active. The polarizabilities
Table 3
MP2 natural orbital occupation numbers larger than 0.00001
obtained at this level are considerably lower than the experimental values. Langhoff et al. [ 121 demon- strated that these low polarizabilities are due to an overcontraction of the active orbitals in the valence CAS calculation. A similar effect has also been no- ticed for other systems, for example [ 22 ] Li-. In the 2~5 1 CT,// [ 20,20,1~~2n;,]~ calculation the antibond- ing 2s orbital has been included in the active space and a weakly occupied orbital has been added for each strong occupied orbital. This does not improve the polarizabilities, as also noted by Jaszuriski et al.
[131.
To increase the active set further it is computa- tionally convenient to employ W constructions.
The 10,10,//[(0)~(2a,1a,1x,)(20*20,2~2~,- l&,)*1 lo calculation incorporates all single and dou- ble excitations from the valence orbitals into a set of virtual orbitals. The virtual set of orbitals in RASIII was initially chosen to contain orbitals with MP2 oc- cupation numbers higher than 0.005. One “0 orbital was then added to ensure additional flexibility. From the occupation numbers of table 3 it is clear that N2 is a multireference system with lx,1 ng defining the principal multireference space. This effect is taken into account in the lo,lo,//[ (20,10,)~( ln,l7c.,)
(20,2a,1x,2~1S,)*] lo calculation (denoted W-L), and this calculation should include all major corre- lation effects. As table 2 shows, the RAS-L calcula- tion gives polarizabilities in close agreement with ex- periment. The agreement with other calculations is also good, a SOPPA calculation at rN_,=2.068 bohr
1.99940 1.98512 1.96343 0.01358 0.00688 0.00101 0.00052 0.00034 o.clOo2o 0.00010 0.00002 0.0000 1 0.0000 I
1.99939 0.12752 3.85731 0.01258 0.00323
1.96711 0.00382 0.01376 0.00053 0.00018
0.02376 0.00125 0.00625
0.00820 0.00034 0.00091
0.00171 0.00009 0.00044
0.00043 0.00003
0.00021 0.00015 0.00011 0.00002 0.00001
Table 4
MCLR frequency-dependent polarizabilities (in au) using the l~~la,//[(2~slu~)4(l~~lr,)(20,2u,l~~2~16,)2]’o contigura- tion space. The numbers in parentheses are experimental values
Frequency (au) (Y Y
0.0 11 58
11:87
(11.76”) 4.65 (4.45 b’)
0.09505 (12.06”) 4.82 (4.89 b))
0.2 12.92 5.42
0.25 13.85 6.02
0.3 15.31 6.83
0.35 17.68 a.22
0.4 22.19 10.77
a) Ref. [19]. b, Ref. 1201.
gave 11.5 7 au for the isotropic component and 4.16 for the anisotropic component [ 143. The MRCI cal-
culation of Langhoff et al. [ 121 gave 11.71 au and 4.62 au for the isotropic and anisotropic compo- nents, respectively, at 2.07432 bohr. For the com- ponents of the static polarlzability we thus have good agreement with the experimental values [ 19,221 as
well as with the best previous ab initio calculations.
The RAS-L calculations had an instability in the ex- citation space of ‘II, symmetry which, however, had an insignificant effect on the polarizabilities. The in- stability is caused by orbital operators connecting or- bitals of similar occupancy and is probably due to a slightly unbalanced choice of active orbital space in one of the irreducible symmetry representations.
The components of the dynamic polarizability were calculated at several frequencies using the RAS-L wavefunction. The results are shown in table 4. At the frequency 0.09505 au (4579 A) we can compare our results, a = 11.87 au, y= 4.82 au with the exper- imental values, cy= 12.06 au, ~~4.89 au. The fre- quency-dependent polarizabilities are therefore also accurately represented by the RAS-L calculation. This is substantiated by comparing the experimental in- dex of refraction (n- 1 + 2niVc~(E) where N is the number of molecules per unit volume) with the in- dex of refraction calculated from the RAS-L polar- izabilities. Fig. 1 shows that the calculated disper- sion of the refractive index is reproduced accurately
8 ? I I I I
7-
2-
I I I I
.I .2 .3 .,G W
Frequency wlau)
Fig. 1. Index of refraction for N2 at 0°C and 1 atm: 0, experimental measurement of ref. [23] ; l , experimental measurement of ref.
[ 241; I, experimental measurement of ref. [ 251;
x ,
calculated with the RAS-L configuration space.Table 5
Polarizabilities of N, (in au) used for numerical differentiation
rN+= 2.068 au rN_N=2.074 au rN_,=2.080au
0~0.0 au fihO.0885585 au w=O.Oau w=O.O885585au o=O.O au okO.0885585 au 30~20Uw
ff 11.4861 11.7047 11.5291 11.7496 11.5722 11.7947
Y 5.1581 5.3278 5.2061 5.3786 5.2543 5.4295
lo*lu”//~~2o*ls~4~ln~l~~~2~~2a”~~2~~~‘~~1’~
01 11.5836 11.8158 11.6216 11.8555 11.6588 11.8943
Y 4.6514 4.7850 4.6917 4.8273 4.7337 4.8713
Table 6
Polarizability derivatives for N, at r,= 2.074 au
3o,2o&,/ /
lqu”//[ (2o,lu,)4(lR,ln”)(2a*2o”l~2~lS,)~]’~
experiment a1
1drld(Rl%) I%-’
o=O.O au w=O.O885585 au
3.19 3.17
3.03 3.09
2.63 f 0.29
lWd(Rl%)l~;’
o=O.O au w=O.O885585 au
1.29 1.32
1.12 1.14
“‘Ref. [ll],
except close to the resonance where Gill and Heddle [ 25 ] report the index of refraction to be 5.7 I x 1 0m4 at 12 16 A. The first resonance occurs at the lowest excitation energy of either ‘Z: or ‘II, symmetry. In the RAS-L calculation these are 0.493 and 0.492 au, respectively. A small error in one of these excitation energies may result in an error in the dispersion as seen in fig. 1. More experimental results would greatly help in judging the quality of the dispersion close to the resonance.
In table 5 we report polarizabilities for N2 in CHF and RAS-L calculations at o=O and w=O.O885585 au. Table 6 gives the polarizability derivatives at
r,= 2.074 au. The results in table 6 show that the po- k&ability derivatives increase slowly with increas- ing frequency of the exciting radiation in agreement with previous SOPPA results of Oddershede and Svendsen [ 261. The derivative of the anisotropy has been measured experimentally by Hamaguchi et al.
[II] at 0=514.5 nm (0.0885585 au) to be 2.43 + 0.29. Based on multireference configuration interaction (MRCI) calculations, Langhoff et al.
[ 121 predict the derivative of the anisotropy to be
3.15 k 0.2. Our RAS calculation supports this prediction.
3. Conclusions
The dynamic dipole polarizability and polariza- bility derivatives of Nz have been studied in the MCLR approximation. In the static limit our results are in good agreement with experiments and other high-accuracy calculations, and this precision is also obtained for the dynamic polarizability, where our accuracy is better than in any previous calculation.
This is very satisfactory since our RAS expansion was designed to include the major correlation effects only.
All calculations reported contain less than 20000 Slater determinants and less than 300 orbital rota- tions. This is well below our current capabilities; we have previously reported calculations containing an order of magnitude more Slater determinants [ 9 1.
The use of MCLR with RAS expansions thus seems to be an economical and accurate way of obtaining dynamic polarizabilities. Calculations on other sys- tems are in progress.
Acknowledgement
This work has been supported by the Danish Nat- ural Science Research Council (Grant No. 1 l-6844).
HJAaJ acknowledges support from the Carlsberg foundation, and TH acknowledges support from the Norwegian Research Council.
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