- Email: [email protected]
Linear response calculations of electronic g-factors and spin-rotational coupling constants for diatomic molecules with a triplet ground state
Chemical Physics 237 Ž1998. 149–158
Linear response calculations of electronic g-factors and spin-rotational coupling constants for diatomic molecules with a triplet ground state ˚ Maria Engstrom, ¨ Boris Minaev, Olav Vahtras, Hans Agren Department of Physics and Measurement Technology, Linkoping UniÕersity, SE-581 83 Linkoping, Sweden ¨ ¨ Received 30 March 1998
Abstract Electronic g-factors for ESR spectra of a number of diatomic molecules with a ground X 3 Sy state and their electronic spin-rotational coupling constants have been calculated by a linear response method. General expressions are used for the second order correction to the electronic g-factor which account for spin-orbit coupling induced admixtures from all excited triplet states to the ground state orbital magnetism. First order corrections – the spin-Zeeman kinetic energy contribution and the one-electron spin-Zeeman gauge contribution – to the g-factor are also accounted for. Calculated g-factors and spin-rotational coupling constants are in a good agreement with available experimental data. In particular, the positive, anomalous, sign of the spin-rotational coupling constant of the PF radical is reproduced. q 1998 Elsevier Science B.V. All rights reserved.
1. Introduction Interactions between paramagnetic compounds and external magnetic fields can be described by an effective spin Hamiltonian w1x H s m B Sg B,
Ž 1.
where m B s e"r2 m e is the Bohr magneton. In this model the g-tensor parametrizes the interaction between the total electronic magnetic moment – or the effective spin S – and the external magnetic field B. The g-shift is dominated by spin-orbit coupling and orbital-Zeeman effects, and contains also several small contributions like the relativistic mass and gauge corrections w1x. Although the knowledge of the magnitude and the anisotropy of g-tensors – that is the deviation from the free electron value – often is indispensable for interpretations of magnetic resonance experiments, comparatively little has so far been accomplished by computational methods. Until recently, g-factors were calculated mostly by semi-empirical methods w2,3x, while lately both density functional ŽDFT. methods w4,5x as well as ab initio techniques including all relevant Breit-Pauli contributions to the g-shift w6x have been developed. A recent note w7x by the present authors described the application of response theory to g-factor calculations. By means of this theory the sum-over-state value for the g-factor is obtained with all intermediate state contributions calculated implicitly. The computational effort of having many 0301-0104r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 1 - 0 1 0 4 Ž 9 8 . 0 0 1 8 8 - 8
M. Engstrom ¨ et al.r Chemical Physics 237 (1998) 149–158
150
excited states computed explicitly and the error in having to truncate the summation over these states are thereby avoided. However, our method is simplified by an approximation where the operator manifold that generates the excited states does not allow for spin polarization. The justification for this approximation is discussed in Ref. w7x, where applications on O 2 and NH including the dominating contributions from the orbital-Zeeman effect and the spin-orbit interaction indicated that this is a viable approach for calculations of g-factors. The purpose of the present study is to further examine the capability of the response theory approach, now also including relativistic mass and one-electron gauge corrections, and to make applications to a series of molecules. Our choice falls on diatomic molecules with a 3 Sy ground state. Spin related properties of these molecules have been studied in several experimental w8–11x and theoretical papers w2,12,13x. Effective spin-spin interactions and singlet-triplet transition intensities for the NH, NF, PH, PF, NCl, O 2 , SO and S 2 molecules were calculated by Wayne and Colbourn w14x. We selected the same sequence of molecules with addition of the two ions OHq and SHq, thereby forming a series with related electronic structure. Furthermore, experimental data of internuclear distances and spin-rotational constants are available for these molecules w15x, making a systematic investigation with experimental comparisons possible.
2. Theory The g-tensor parametrizes the interaction between magnetic moments and external magnetic fields, and is often reported as a correction of the free electron value, g e s 2.002319 g s g e 1 q D g.
Ž 2.
A detailed description of the formalism used in this work can be found in Harriman w1x. Several terms contribute to the g-shift D g. These are obtained from a perturbation expansion of the molecular energy, retaining terms bilinear in electron spin and magnetic field intensity. D g a b s D g RM C q D g GC Ž 1e . q D g GC Ž 2 e . q D g SOCqOZ Ž 1e . q D g SOCqOZ Ž 2 e . .
Ž 3.
The g-shift is dominated by the second order spin-orbit coupling and orbital-Zeeman one- and two-electron cross terms Ž D g SOCqOZ Ž1e .,D g SOCqOZ Ž2 e ... First order relativistic mass Ž D g RMC . and gauge corrections Ž D g GC Ž1e .,D g GC Ž2 e .. provide additional contributions. The isotropic relativistic mass contribution can be described by ab D g RM Cs
a2 S
²C 0 < Ý Pi2 s z ,i
Ž 4.
i
where the sum runs over all electrons i, d a b is the Kronecker delta symbol, a s 7.297353 = 10y3 is the fine structure constant, and P is the linear momentum. We assume that C 0 refers to the ground state component with maximum spin projection M s s S. Furthermore, we do not make distinction between g e and 2 in higher-order expressions. Calculations of g-tensors depend on the origin of the magnetic vector potential A which enters the kinetic momentum
P s p q eA ,
Ž 5.
where p is the canonical momentum. For a homogeneous magnetic field, the vector potential is normally chosen as A s 12 B = r C
Ž 6.
M. Engstrom ¨ et al.r Chemical Physics 237 (1998) 149–158
151
defined with respect to an arbitrary gauge origin r C , which in this work is chosen as the electronic charge centroid ŽECC. defined by rC s
1 N
N
Ý ri .
Ž 7.
is1
Substituting this into the spin-orbit operator yields the one- and two-electron gauge corrections. ab D g GC Ž 1e . s
ab D g GC Ž2 e. s
a2 4S
a2 4S
²C 0 < Ý i
²C 0 < Ý i
Z
K Ý r 3 Ž r i K P r iC . d a b y r iCa r ibK
K
1
Ý j/i
s z ,i
Ž 8.
iK
a a r i j P Ž 2 r jC y r iC . d a b y Ž 2 r jC y r iC . r ibj s z , j
ri3j
Ž 9.
where the sums run over all electrons i, j and nuclear centers K. Previous calculations w16,6x showed that the choice of gauge origin in general has a modest effect on the electronic g-factor if the basis set includes polarization functions. In the present work a 4 s3 p2 d1 f contracted basis is used for all atoms except hydrogen and the formalism is thereby expected to be nearly gauge invariant. Second order contributions involve the spin-orbit coupling and orbital Zeeman operators. ab D g SOCqOZ Ž 1e . s
²C 0 < Ý
a2 S
ab D g SOCqOZ Ž2 e. sy
i
Ý
Ý Ž ZKrri3K . l iaK s z ,i
i
n
²C 0 < Ý
a2 S
,
E 0 y En i
Ý n
Ž 10 .
Ý Ž l iajrri3j q 2 l jia rri3j . s z ,i
i
E 0 y En
,
Ž 11 .
where the sum is over excited states Cn of same spin as C 0 and l i K ,l iC , and l i j denote angular momentum elements. It is not possible to observe the anisotropic part of the g-tensor for molecules in the gas phase. However, the electron spin-rotational tensor which couples the spin and the rotational angular momentum of molecules is observable in the gas phase. Curl w17x derived an approximate relation between the electronic spin-rotational coupling constants and the g-tensor,
g 0 s y2 Be D g H ,
Ž 12 .
where D g H is the transversal component of the g-shift. Be the rotational constant Be s " 2r2 I
Ž 13 .
and I the moment of inertia. We have used Curl’s formula Ž12. for the estimation of the electronic spin-rotational coupling constants since these values are available from analysis of rotational structure of electronic transitions in the visible and UV regions and from microwave spectroscopy w15x. All studied molecules appear to conform to the Hund’s case Žb. so the electronic spin-rotational coupling constants are easily determined from spectra. Only the transversal components g H s g x x s g y y need to be calculated in diatomic molecules with a 3 Sy ground state. Contributions of the second order perturbation term from orbital magnetism to g 5 is zero due to symmetry, and only small first order corrections Žspin-Zeeman kinetic energy or relativistic mass contribution and spin-orbit gauge contribution. give nonzero values. The g-factors have been measured for two molecules of this series, namely O 2 and SO w18x. Deviations from the free electron value for g 5 are of the order of 2 = 10y4 which could be accounted for by the relativistic mass correction. By these reasons we will not consider the g 5 values.
M. Engstrom ¨ et al.r Chemical Physics 237 (1998) 149–158
152
3. Methods Transversal g-factors g H were calculated with the DALTON quantum chemistry program w19x. Calculated g-shifts include the following corrections D g a b s D g RM C d a b q D g GC Ž 1e . q D g SOCqOZ Ž 1e . q D g SOCqOZ Ž 2 e . .
Ž 14 .
The two-electron gauge correction is not included in this work. Spin rotational constants g 0 were calculated according to Curl’s relation Ž12.. Experimental values of internuclear distances were obtained from Huber and Herzberg w15x. We have used multi-electron self-consistent field wave functions ŽMCSCF. with correlating complete active spaces. Results are reported for two active spaces, CAS1 and CAS2, which in C 2 Õ symmetry notation Ž A1 B1 B2 A 2 . read 4220 and 6331, respectively. The NH, PH, OHq and SHq molecules were calculated with all 6 valence electrons in the active space, while the remaining molecules were assigned 10 active electrons, leaving out only core and inner-most valence electrons. In order to make estimations of electron correlation effects we constructed open shell restricted Hartree-Fock ŽORHF. like wave functions with two electrons in a small p-orbital active space Ž0110.. We have used experimental values for the internuclear distances in order to properly examine the capacity for g-factor calculations. The Atomic Natural Orbitals ŽANO. basis sets of Widmark et al. w20x were used in all calculations. The lowest recommended contraction level for correlated wave functions was then employed. This implies a 3s2 p1d contraction for hydrogen atom and a 4 s3 p2 d1 f contraction for nitrogen, oxygen, fluorine, phosphorus, sulfur and chlorine atoms.
4. Result and discussion Table 1 shows the different terms contributing to the g-shift; the relativistic mass contribution Ž D g RM C ., the one-electron gauge correction Ž D g GC Ž1e .. and the spin-orbitrorbital-Zeeman Ž D g SOCqOZ . contribution. The largest contribution originates from the spin-orbitrorbital-Zeeman effect. The two-electron gauge correction, not considered here, is expected to be of the same magnitude as the one-electron correction, but with opposite sign, w12x and Christensen w21x et al., see Table 2. The results indicate that one- and as predicted by Bundgen ¨ two-electron corrections almost cancel each other. The effect of electron correlation on first order corrections are small compared to the total g-shift. The largest variation among first order terms is less than 2 percent of the
Table 1 Contributions to the g-shift. The table provides a comparison of calculations with three different methods described in the text. D g RM C is the relativistic mass contribution and D g GC Ž1e . is the one electron gauge correction. D g SOCqOZ is the second order spin-orbitrorbital-Zeeman contribution. All values are given in ppm Molecule
O2 SO S2 NH NF NCl PH PF OHq SHq
D g RM C
D g GC Ž1e .
D g SOCqOZ
ORHF
CAS1
CAS2
ORHF
CAS1
CAS2
ORHF
CAS1
CAS2
y322. y232. y211. y192. y238. y219. y145. y169. y305. y220.
y326. y254. y216. y212. y253. y221. y151. y177. y335. y227.
y367. y259. y222. y206. y271. y240. y148. y184. y328. y226.
208. 254. 251. 106. 133. 142. 137. 158. 152. 176.
204. 235. 239. 111. 137. 142. 139. 157. 160. 168.
216. 255. 263. 109. 143. 174. 136. 161. 157. 168.
3470. 4321. 12622. 1637. 1413. 4026. 4090. y1251. 4623. 8178.
2774. 4187. 11424. 1537. 1897. 4226. 4506. y2679. 4154. 8744.
2957. 4068. 11177. 1536. 2056. 4141. 4045. y602. 4197. 7901.
M. Engstrom ¨ et al.r Chemical Physics 237 (1998) 149–158
153
Table 2 First order contributions to the g-shift for the O 2 and SO molecules. D g RMC is the relativistic mass contribution. D g GC Ž1e . and D g GC Ž1e . are the one- and two-electron gauge corrections, respectively. All values are given in ppm
O2
SO
a b
a
Christensen , SCF b Bundgen , SCF ¨ b Bundgen , CI ¨ This work, ORHF This work, CAS1 This work, CAS2 Christensen, SCF Bundgen, SCF ¨ Bundgen, CI ¨ This work, ORHF This work, CAS1 This work, CAS2
D g RM C
D g GC Ž1e .
D g GC Ž2 e .
y322 y322 y355 y322 y326 y367 y242 y234 y259 y232 y254 y259
218 218 229 208 204 216 237 235 242 254 235 255
y151 y173
y208 y202
Ref. w21x. Ref. w12x.
total correction. Second order contributions are more sensitive to electron correlation. The difference between ORHF and MCSCF second order contributions makes a considerable portion of the total g-shift, while the deviation of CAS1 from CAS2 results is less significant. w12x and Christensen w21x calculated first order contributions to the g-shift for O 2 and SO. In Table Bundgen ¨ 2 we present their results together with calculations of this work. A comparison of these results confirms the conclusion that electron correlation is of less importance to first order corrections. This observation allow us to use the two-electron result by Christensen or Bundgen as additional corrections to our calculations for O 2 and ¨ SO. Table 3 shows the total correction of the transversal g-values D g H . One trend is recognized by comparing the isovalent molecules. The g-shifts increase when going from lower to higher row of atoms. That is, D g H ŽNH. - D g H ŽPH., D g H ŽNF. - D g H ŽNCl. and D g H ŽO 2 . - D g H ŽSO. - D g H ŽS 2 .. The reason for this is that spin-orbit coupling increases with larger nuclear charge. Calculations w22x of vertical energies for O 2 , SO, and S 2 show that the energy gap D E between the mostly contributing 3 P states and the 3 S ground state decreases with the increase of nuclear charge. Larger spin-orbit coupling and a smaller energy gap will together Table 3 Total correction to g-factors. D g H values of diatomic molecules with the ground triplet 3 Sy state. The g-shifts are given in ppm D gH Molecule
˚. Distance ŽA
O2 SO S2 NH NF NCl PH PF OHq SHq
1.207 1.481 1.889 1.036 1.317 1.614 1.422 1.590 1.029 1.374
a
Ref. w15x.
a
ORHF
CAS1
CAS2
3355. 4343. 12662. 1552. 1308 3949. 4082. y1263. 4470. 8134.
2652. 4168. 11448. 1436. 1781. 4147. 4494. y2698. 3980. 8686.
2806. 4064. 11218. 1438. 1928. 4074. 4033. y625. 4026. 7842.
154
M. Engstrom ¨ et al.r Chemical Physics 237 (1998) 149–158
Fig. 1. g-shift as a function of internuclear distance in the NH molecule.
contribute to a larger g-shift. Cross comparisons of isovalent neutral and ionic molecules show the same trend: D g H ŽNH. - D g H ŽOHq. and D g H ŽPH. - D g H ŽSHq. . Going from the neutral to the ionic molecule the nuclear charge will increase by one unit and thereby increase the spin-orbit coupling. SOC will also increase for the same nucleus when going from the neutral atom to its ion. D E is also slightly smaller for OHq than NH w15x and this energy difference may provide additional interpretation of the higher g-shifts for the ions. If the assumption of a negative two-electron gauge correction is correct, then the calculated g-shift is slightly magnified. We made a further correction of the g-shift for O 2 at the CAS1 level by adding the two-electron term from Christensen. This additional correction gives g H s 2.004820, which is in very good agreement with the experimental value, g H s 2.004838, deduced by Bowers et al. w23x. The g-shifts for NH and S 2 are plotted as a function of internuclear distance in Figs. 1 and 2. A similar performance has been obtained in the previous work on NH and O 2 molecules w7x, where the first order corrections were not accounted for. The g-shift for NH is decreasing with internuclear distance while the opposite phenomenon occurs for the O 2 w7x and S 2 molecules ŽFigs. 1,2.. The individual contributions to the g-shift of the NH molecule are plotted in Fig. 3. First order contributions are almost independent of internuclear distance. The distance dependence of the g-shift is attributed to second order terms. The most important contribution to the second order term originates from the SOC mixing between the ground 3 Sy state and the
Fig. 2. g-shift as a function of internuclear distance in the S 2 molecule.
M. Engstrom ¨ et al.r Chemical Physics 237 (1998) 149–158
155
Fig. 3. Individual contributions to g-shift as a function of interatomic distance in the NH molecule. The solid line shows the second order spin-orbitrorbital-Zeeman contribution. The dashed lines represents first order one-electron gauge correction Ž-P. and relativistic mass contribution Ž – ., respectively.
first excited 3 P state Žthe 3 P g state in the S 2 case.. The SOC integral between these states and the orbital angular momentum matrix elements are not strong functions of the internuclear distance for the O 2 and S 2 3 molecules. The energy difference between the ground 3 Sy g state and the first excited P g states is therefore the crucial value. The energy of these two states are plotted as a function of internuclear distance in Figs. 4 and 5. These plots together with Eqs. Ž10. and Ž11. explain the opposite performance of the NH and S 2 g-factors. In S 2 both states refer to the same dissociation limit with nonzero SOC constant. In NH the ground 3 Sy state and the first excited 3 P state refer to different dissociation limits Ž 2 P and 4 S states of nitrogen atom, respectively. with a large energy difference. The excitation energy at r s 2.6 au has slowly decreased to 3.69 eV from 3.82 at the equilibrium distance. The angular momentum is almost constant in the interval 1.6 to 2.5 au ŽFig. 6.. The spin-orbit coupling with the 13 P state is on the other hand decreasing in this interval ŽFig. 7.. No other states Ž n G 2. contribute much to the g-factor since they have small orbital angular momentum matrix elements with the ground state and large energy denominators. Spin-orbit coupling with the lowest 3 P state is thereby the most important effect describing the reduction of the g-shift with increasing intermolecular distance in the NH molecule. A similar performance of the g-factor and spin-rotational coupling constant can be predicted for the isoelectronic PH, OHq and SHq radicals.
Fig. 4. The energy of the ground 3 Sy state and the first excited 3 P state as a function of internuclear distance in the NH molecule.
156
M. Engstrom ¨ et al.r Chemical Physics 237 (1998) 149–158
Fig. 5. The energy of the ground 3 Sy state and the first excited 3 P state as a function of internuclear distance in the S 2 molecule.
Fig. 6. Angular momentum integrals. The integrals are calculated for the six lowest excited 3 P states as a function of interatomic distance in ˚ the NH molecule. Equilibrium distance is 1.9 A.
Fig. 7. Spin-orbit coupling integrals. The integrals are calculated for the six lowest excited 3 P states as a function of interatomic distance in ˚ NH molecule. Equilibrium distance is 1.9 A.
M. Engstrom ¨ et al.r Chemical Physics 237 (1998) 149–158
157
Table 4 Comparisons with experiments. Rotational Be and Spin rotational g 0 constants of diatomic molecules with the ground triplet 3 Sy state Molecule
Be Žcmy1 .
g 0 Ž10y3 cmy1 . Experiment
O2 SO S2 NH NF NCl PH PF OHq SHq a
1.445 0.721 0.295 16.702 1.205 0.647 8.536 0.567 16.802 9.136
a
y8.42536 y5.6153 y6.653 y54.66 y4.8 y7.15 y73.8 q1.8 y147.8 y165
This work, CAS1
This work, CAS2
y7.67 y6.01 y6.77 y48.0 y4.3 y5.4 y76.7 q3.1 y133.7 y158.7
y8.12 y5.86 y6.63 y48.0 y4.6 y5.3 y68.9 q0.7 y135.3 y143.3
Huber and Herzberg w15x.
All calculated g-shifts are positive except for PF, for which the correction is negative. The negative sign of the g-factor of PF is in agreement with experiment Žsee Table 4.. One aim of this study was to estimate if there was a significant difference between the results obtained with restricted Hartree-Fock Žwith only one configuration state function. at one hand and the correlated many-configuration ŽCAS1 and CAS2. results on the other. One finds that MCSCF wave functions are preferred since the disparity between ORHF and correlated methods can amount to a quarter of the total g-shift. On the other hand the difference between CAS1 and CAS2 is smaller. Regarding the increase in computational effort by enlarging the active space from CAS1 to CAS2, one can recommend an active space of moderate size. Experimental and calculated values for spin-rotational g 0 and rotational constants Be are tabulated in Table 4. The calculations show a reasonable agreement with experiment. In comparing calculated and experimental data there are two facts to be reflected upon. First, the Curl relation is an approximation to the exact spin rotational constant, and the influence of this approximation upon our results is somewhat hard to estimate. Second, a deviation in the first decimal of the g 0 value can be expected, due to the fact that the two-electron gauge correction is not included. Using the two-electron correction for O 2 and SO yields g 0 s y7.23024 = 10y3 cmy1 and g 0 s y5.7082 = 10y3 cmy1 , respectively. That is, the experimental agreement will be improved for SO but slightly deteriorated for O 2 . The calculated internuclear distance dependence of the g-factor leads to the same type of dependence for the spin-rotational coupling constant through Curl’s formula Ž10.. It means that the absolute value of the ge constant will increase with the internuclear distance and with the vibrational quantum number for O 2 , SO and S 2 but will decrease for the NH molecule. These trends are well reproduced by experimental findings w15x. The spin-rotational coupling constant for the first excited vibrational quantum state Ž Õ s 1. in NH is determined to be g 1 s y51.7 = 10y3 cmy1. The absolute value is about 6 percent smaller than the ground state Ž Õ s 0. value, g 0 s y54.66 = 10y3 cmy1. In S 2 , g 1 s y6.716 = 10y3 cmy1 , i.e an absolute value higher than for the ground state, g 0 s y6.653 = 10y3 cmy1 . The same trend is seen for O 2 and SO Žg 1 s y8.4468 and y5.72 = 10y3 cmy1 respectively. w15x. Accounting for the approximate nature of Curl’s formula we can say that agreement with experimental data for the electronic spin-rotational coupling constants is good. All qualitative trends are well reproduced. The most puzzling feature is a positive sign of the spin-rotational coupling constant for the PF radical. We have reproduced this sign though we have not obtained a simple explanation of this feature in terms of individual contributions. Analysis of contributions from the ten lowest excited states does not reveal the reason for this anomaly, and apparently some higher excited states are responsible for a delicate balance between positive and negative contributions to the spin-rotational coupling constant in the PF radical. This observation points at the problem with truncations of the sum-over state value, since such a truncation can break the delicate balance
M. Engstrom ¨ et al.r Chemical Physics 237 (1998) 149–158
158
between different contributing states. An account of electronic correlation can also be important for this balance; we note the experimental value for PF is just in the middle between the CAS1 and CAS2 results.
5. Conclusions In the present work we have explored the applicability of a linear response method for the calculations of electronic g-factors and spin-rotational coupling constants. We have used a series of diatomic molecules with a triplet ground state and with similar electronic structures for this purpose. Overall, we find results in good agreement with experimental data, concerning both the trends and the magnitudes of the total values. In particular, the anomalous sign and the magnitude for one member, PF, in the series could be reproduced. Correlated wave functions are essential for accurate calculations of the second order contributions to the g-shift, while the first order corrections are less sensitive. In some cases the calculations indicated the need to include the complete summation of the excited-state contributions, since upper excited states may provide a delicate balance in sign and magnitude of their different contributions.
Acknowledgements This work was supported by the INTAS grant 94-4089 and by the Swedish Foundation for Strategic Research. Computer time has been supported by NSC, the Swedish national supercomputer centre in Linkoping. ¨
References w1x w2x w3x w4x w5x w6x w7x w8x w9x w10x w11x w12x w13x w14x w15x w16x w17x w18x w19x
w20x w21x w22x w23x
J.E. Harriman, Theoretical foundations of electron spin resonance, Academic Press, New York, 1978. B.F. Minaev, J. Mol. Struct. ŽTheochem. 183 Ž1989. 207. M. Ishii, K. Morihashi, O. Kikuchi, J. Mol. Struct. ŽTheochem. 187 Ž1989. 209. H.G. Schreckenbach, Relativity and magnetic properties. A density functional study, Ph.D. thesis, University of Calgary, Calgary, Alberta, Canada, 1996. E. van Lenthe, P.E.S. Wormer, A. van der Avoird, J. Chem. Phys. 107 Ž1997. 2488. G.H. Lushington, F. Grein, J. Chem. Phys. 106 Ž1997. 3292. ˚ O. Vahtras, B. Minaev, H. Agren, Chem. Phys. Letters 281 Ž1997. 186. M.H.W. Gruebele, R.P. Muller, R.J. Saykally, J. Chem. Phys. 84 Ž1986. 2489. ¨ K. Kawaguchi, C. Yamada, E. Hirota, J. Chem. Phys. 71 Ž1979. 3338. L. Tomuta, M. Mizushima, C.J. Howard, K.M. Evenson, Phys. Rev. 12 Ž1975. 974. F.D. Wayne, H.E. Radford, Mol. Phys. 32 Ž1976. 1407. P. Bundgen, G. Lushington, F. Grein, Int. J. Quant. Chem. Symp. 29 Ž1995. 283. ¨ M. Tinkham, M.W.P. Strandberg, Phys. Rev. 97 Ž1955. 951. F.D. Wayne, E.A. Colbourn, Mol. Phys. 34 Ž1977. 1141. K.P. Huber, G. Herzberg, Molecular spectra and molecular structure. Constants of diatomic molecules, Van Nostrand Reinhold Company, New York, 1979. G.H. Lushington, P. Bundgen, F. Grein, Int. J. Quant. Chem. 55 Ž1995. 377. ¨ R.F. Curl, Mol. Phys. 9 Ž1965. 565. W. Weltner Jr., Magnetic Atoms and Molecules, Dover, New York, 1987. ˚ T. Helgaker, H.J.Aa. Jensen, P. Jørgensen, J. Olsen, K. Ruud, H. Agren, T. Andersen, K.L. Bak, V. Bakken, O. Christiansen, P. Dahle, E.K. Dalskov, T. Enevoldsen, B. Fernandez, H. Heiberg, H. Hettema, D. Jonsson, S. Kirpekar, R. Kobayashi, H. Koch, K.V. Mikkelsen, P. Norman, M.J. Packer, T. Saue, P.R. Taylor, O. Vahtras, DALTON, a second-order MCSCF molecular property program. ˚ Malmquist, B.O. Roos, Theor. Chim. Acta 77 Ž1990. 291. P-O. Widmark, P.A. H. Christensen, L. Veseth, J. Mol. Spect. 72 Ž1978. 438. R. Klotz, Ch.M. Marian, S.D. Peyerimhoff, Chem. Phys. 89 Ž1984. 223. K.D. Bowers, R.A. Kamper, C.D. Lustig, Proc. Roy. Soc. London. A 251 Ž1959. 565.
Linear response calculations of electronic g-factors and spin-rotational coupling constants for diatomic molecules with a triplet ground state ˚ Maria Engstrom, ¨ Boris Minaev, Olav Vahtras, Hans Agren Department of Physics and Measurement Technology, Linkoping UniÕersity, SE-581 83 Linkoping, Sweden ¨ ¨ Received 30 March 1998
Abstract Electronic g-factors for ESR spectra of a number of diatomic molecules with a ground X 3 Sy state and their electronic spin-rotational coupling constants have been calculated by a linear response method. General expressions are used for the second order correction to the electronic g-factor which account for spin-orbit coupling induced admixtures from all excited triplet states to the ground state orbital magnetism. First order corrections – the spin-Zeeman kinetic energy contribution and the one-electron spin-Zeeman gauge contribution – to the g-factor are also accounted for. Calculated g-factors and spin-rotational coupling constants are in a good agreement with available experimental data. In particular, the positive, anomalous, sign of the spin-rotational coupling constant of the PF radical is reproduced. q 1998 Elsevier Science B.V. All rights reserved.
1. Introduction Interactions between paramagnetic compounds and external magnetic fields can be described by an effective spin Hamiltonian w1x H s m B Sg B,
Ž 1.
where m B s e"r2 m e is the Bohr magneton. In this model the g-tensor parametrizes the interaction between the total electronic magnetic moment – or the effective spin S – and the external magnetic field B. The g-shift is dominated by spin-orbit coupling and orbital-Zeeman effects, and contains also several small contributions like the relativistic mass and gauge corrections w1x. Although the knowledge of the magnitude and the anisotropy of g-tensors – that is the deviation from the free electron value – often is indispensable for interpretations of magnetic resonance experiments, comparatively little has so far been accomplished by computational methods. Until recently, g-factors were calculated mostly by semi-empirical methods w2,3x, while lately both density functional ŽDFT. methods w4,5x as well as ab initio techniques including all relevant Breit-Pauli contributions to the g-shift w6x have been developed. A recent note w7x by the present authors described the application of response theory to g-factor calculations. By means of this theory the sum-over-state value for the g-factor is obtained with all intermediate state contributions calculated implicitly. The computational effort of having many 0301-0104r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 1 - 0 1 0 4 Ž 9 8 . 0 0 1 8 8 - 8
M. Engstrom ¨ et al.r Chemical Physics 237 (1998) 149–158
150
excited states computed explicitly and the error in having to truncate the summation over these states are thereby avoided. However, our method is simplified by an approximation where the operator manifold that generates the excited states does not allow for spin polarization. The justification for this approximation is discussed in Ref. w7x, where applications on O 2 and NH including the dominating contributions from the orbital-Zeeman effect and the spin-orbit interaction indicated that this is a viable approach for calculations of g-factors. The purpose of the present study is to further examine the capability of the response theory approach, now also including relativistic mass and one-electron gauge corrections, and to make applications to a series of molecules. Our choice falls on diatomic molecules with a 3 Sy ground state. Spin related properties of these molecules have been studied in several experimental w8–11x and theoretical papers w2,12,13x. Effective spin-spin interactions and singlet-triplet transition intensities for the NH, NF, PH, PF, NCl, O 2 , SO and S 2 molecules were calculated by Wayne and Colbourn w14x. We selected the same sequence of molecules with addition of the two ions OHq and SHq, thereby forming a series with related electronic structure. Furthermore, experimental data of internuclear distances and spin-rotational constants are available for these molecules w15x, making a systematic investigation with experimental comparisons possible.
2. Theory The g-tensor parametrizes the interaction between magnetic moments and external magnetic fields, and is often reported as a correction of the free electron value, g e s 2.002319 g s g e 1 q D g.
Ž 2.
A detailed description of the formalism used in this work can be found in Harriman w1x. Several terms contribute to the g-shift D g. These are obtained from a perturbation expansion of the molecular energy, retaining terms bilinear in electron spin and magnetic field intensity. D g a b s D g RM C q D g GC Ž 1e . q D g GC Ž 2 e . q D g SOCqOZ Ž 1e . q D g SOCqOZ Ž 2 e . .
Ž 3.
The g-shift is dominated by the second order spin-orbit coupling and orbital-Zeeman one- and two-electron cross terms Ž D g SOCqOZ Ž1e .,D g SOCqOZ Ž2 e ... First order relativistic mass Ž D g RMC . and gauge corrections Ž D g GC Ž1e .,D g GC Ž2 e .. provide additional contributions. The isotropic relativistic mass contribution can be described by ab D g RM Cs
a2 S
²C 0 < Ý Pi2 s z ,i
Ž 4.
i
where the sum runs over all electrons i, d a b is the Kronecker delta symbol, a s 7.297353 = 10y3 is the fine structure constant, and P is the linear momentum. We assume that C 0 refers to the ground state component with maximum spin projection M s s S. Furthermore, we do not make distinction between g e and 2 in higher-order expressions. Calculations of g-tensors depend on the origin of the magnetic vector potential A which enters the kinetic momentum
P s p q eA ,
Ž 5.
where p is the canonical momentum. For a homogeneous magnetic field, the vector potential is normally chosen as A s 12 B = r C
Ž 6.
M. Engstrom ¨ et al.r Chemical Physics 237 (1998) 149–158
151
defined with respect to an arbitrary gauge origin r C , which in this work is chosen as the electronic charge centroid ŽECC. defined by rC s
1 N
N
Ý ri .
Ž 7.
is1
Substituting this into the spin-orbit operator yields the one- and two-electron gauge corrections. ab D g GC Ž 1e . s
ab D g GC Ž2 e. s
a2 4S
a2 4S
²C 0 < Ý i
²C 0 < Ý i
Z
K Ý r 3 Ž r i K P r iC . d a b y r iCa r ibK
K
1
Ý j/i
s z ,i
Ž 8.
iK
a a r i j P Ž 2 r jC y r iC . d a b y Ž 2 r jC y r iC . r ibj s z , j
ri3j
Ž 9.
where the sums run over all electrons i, j and nuclear centers K. Previous calculations w16,6x showed that the choice of gauge origin in general has a modest effect on the electronic g-factor if the basis set includes polarization functions. In the present work a 4 s3 p2 d1 f contracted basis is used for all atoms except hydrogen and the formalism is thereby expected to be nearly gauge invariant. Second order contributions involve the spin-orbit coupling and orbital Zeeman operators. ab D g SOCqOZ Ž 1e . s
²C 0 < Ý
a2 S
ab D g SOCqOZ Ž2 e. sy
i
Ý
Ý Ž ZKrri3K . l iaK s z ,i
i
n
²C 0 < Ý
a2 S
,
E 0 y En i
Ý n
Ž 10 .
Ý Ž l iajrri3j q 2 l jia rri3j . s z ,i
i
E 0 y En
,
Ž 11 .
where the sum is over excited states Cn of same spin as C 0 and l i K ,l iC , and l i j denote angular momentum elements. It is not possible to observe the anisotropic part of the g-tensor for molecules in the gas phase. However, the electron spin-rotational tensor which couples the spin and the rotational angular momentum of molecules is observable in the gas phase. Curl w17x derived an approximate relation between the electronic spin-rotational coupling constants and the g-tensor,
g 0 s y2 Be D g H ,
Ž 12 .
where D g H is the transversal component of the g-shift. Be the rotational constant Be s " 2r2 I
Ž 13 .
and I the moment of inertia. We have used Curl’s formula Ž12. for the estimation of the electronic spin-rotational coupling constants since these values are available from analysis of rotational structure of electronic transitions in the visible and UV regions and from microwave spectroscopy w15x. All studied molecules appear to conform to the Hund’s case Žb. so the electronic spin-rotational coupling constants are easily determined from spectra. Only the transversal components g H s g x x s g y y need to be calculated in diatomic molecules with a 3 Sy ground state. Contributions of the second order perturbation term from orbital magnetism to g 5 is zero due to symmetry, and only small first order corrections Žspin-Zeeman kinetic energy or relativistic mass contribution and spin-orbit gauge contribution. give nonzero values. The g-factors have been measured for two molecules of this series, namely O 2 and SO w18x. Deviations from the free electron value for g 5 are of the order of 2 = 10y4 which could be accounted for by the relativistic mass correction. By these reasons we will not consider the g 5 values.
M. Engstrom ¨ et al.r Chemical Physics 237 (1998) 149–158
152
3. Methods Transversal g-factors g H were calculated with the DALTON quantum chemistry program w19x. Calculated g-shifts include the following corrections D g a b s D g RM C d a b q D g GC Ž 1e . q D g SOCqOZ Ž 1e . q D g SOCqOZ Ž 2 e . .
Ž 14 .
The two-electron gauge correction is not included in this work. Spin rotational constants g 0 were calculated according to Curl’s relation Ž12.. Experimental values of internuclear distances were obtained from Huber and Herzberg w15x. We have used multi-electron self-consistent field wave functions ŽMCSCF. with correlating complete active spaces. Results are reported for two active spaces, CAS1 and CAS2, which in C 2 Õ symmetry notation Ž A1 B1 B2 A 2 . read 4220 and 6331, respectively. The NH, PH, OHq and SHq molecules were calculated with all 6 valence electrons in the active space, while the remaining molecules were assigned 10 active electrons, leaving out only core and inner-most valence electrons. In order to make estimations of electron correlation effects we constructed open shell restricted Hartree-Fock ŽORHF. like wave functions with two electrons in a small p-orbital active space Ž0110.. We have used experimental values for the internuclear distances in order to properly examine the capacity for g-factor calculations. The Atomic Natural Orbitals ŽANO. basis sets of Widmark et al. w20x were used in all calculations. The lowest recommended contraction level for correlated wave functions was then employed. This implies a 3s2 p1d contraction for hydrogen atom and a 4 s3 p2 d1 f contraction for nitrogen, oxygen, fluorine, phosphorus, sulfur and chlorine atoms.
4. Result and discussion Table 1 shows the different terms contributing to the g-shift; the relativistic mass contribution Ž D g RM C ., the one-electron gauge correction Ž D g GC Ž1e .. and the spin-orbitrorbital-Zeeman Ž D g SOCqOZ . contribution. The largest contribution originates from the spin-orbitrorbital-Zeeman effect. The two-electron gauge correction, not considered here, is expected to be of the same magnitude as the one-electron correction, but with opposite sign, w12x and Christensen w21x et al., see Table 2. The results indicate that one- and as predicted by Bundgen ¨ two-electron corrections almost cancel each other. The effect of electron correlation on first order corrections are small compared to the total g-shift. The largest variation among first order terms is less than 2 percent of the
Table 1 Contributions to the g-shift. The table provides a comparison of calculations with three different methods described in the text. D g RM C is the relativistic mass contribution and D g GC Ž1e . is the one electron gauge correction. D g SOCqOZ is the second order spin-orbitrorbital-Zeeman contribution. All values are given in ppm Molecule
O2 SO S2 NH NF NCl PH PF OHq SHq
D g RM C
D g GC Ž1e .
D g SOCqOZ
ORHF
CAS1
CAS2
ORHF
CAS1
CAS2
ORHF
CAS1
CAS2
y322. y232. y211. y192. y238. y219. y145. y169. y305. y220.
y326. y254. y216. y212. y253. y221. y151. y177. y335. y227.
y367. y259. y222. y206. y271. y240. y148. y184. y328. y226.
208. 254. 251. 106. 133. 142. 137. 158. 152. 176.
204. 235. 239. 111. 137. 142. 139. 157. 160. 168.
216. 255. 263. 109. 143. 174. 136. 161. 157. 168.
3470. 4321. 12622. 1637. 1413. 4026. 4090. y1251. 4623. 8178.
2774. 4187. 11424. 1537. 1897. 4226. 4506. y2679. 4154. 8744.
2957. 4068. 11177. 1536. 2056. 4141. 4045. y602. 4197. 7901.
M. Engstrom ¨ et al.r Chemical Physics 237 (1998) 149–158
153
Table 2 First order contributions to the g-shift for the O 2 and SO molecules. D g RMC is the relativistic mass contribution. D g GC Ž1e . and D g GC Ž1e . are the one- and two-electron gauge corrections, respectively. All values are given in ppm
O2
SO
a b
a
Christensen , SCF b Bundgen , SCF ¨ b Bundgen , CI ¨ This work, ORHF This work, CAS1 This work, CAS2 Christensen, SCF Bundgen, SCF ¨ Bundgen, CI ¨ This work, ORHF This work, CAS1 This work, CAS2
D g RM C
D g GC Ž1e .
D g GC Ž2 e .
y322 y322 y355 y322 y326 y367 y242 y234 y259 y232 y254 y259
218 218 229 208 204 216 237 235 242 254 235 255
y151 y173
y208 y202
Ref. w21x. Ref. w12x.
total correction. Second order contributions are more sensitive to electron correlation. The difference between ORHF and MCSCF second order contributions makes a considerable portion of the total g-shift, while the deviation of CAS1 from CAS2 results is less significant. w12x and Christensen w21x calculated first order contributions to the g-shift for O 2 and SO. In Table Bundgen ¨ 2 we present their results together with calculations of this work. A comparison of these results confirms the conclusion that electron correlation is of less importance to first order corrections. This observation allow us to use the two-electron result by Christensen or Bundgen as additional corrections to our calculations for O 2 and ¨ SO. Table 3 shows the total correction of the transversal g-values D g H . One trend is recognized by comparing the isovalent molecules. The g-shifts increase when going from lower to higher row of atoms. That is, D g H ŽNH. - D g H ŽPH., D g H ŽNF. - D g H ŽNCl. and D g H ŽO 2 . - D g H ŽSO. - D g H ŽS 2 .. The reason for this is that spin-orbit coupling increases with larger nuclear charge. Calculations w22x of vertical energies for O 2 , SO, and S 2 show that the energy gap D E between the mostly contributing 3 P states and the 3 S ground state decreases with the increase of nuclear charge. Larger spin-orbit coupling and a smaller energy gap will together Table 3 Total correction to g-factors. D g H values of diatomic molecules with the ground triplet 3 Sy state. The g-shifts are given in ppm D gH Molecule
˚. Distance ŽA
O2 SO S2 NH NF NCl PH PF OHq SHq
1.207 1.481 1.889 1.036 1.317 1.614 1.422 1.590 1.029 1.374
a
Ref. w15x.
a
ORHF
CAS1
CAS2
3355. 4343. 12662. 1552. 1308 3949. 4082. y1263. 4470. 8134.
2652. 4168. 11448. 1436. 1781. 4147. 4494. y2698. 3980. 8686.
2806. 4064. 11218. 1438. 1928. 4074. 4033. y625. 4026. 7842.
154
M. Engstrom ¨ et al.r Chemical Physics 237 (1998) 149–158
Fig. 1. g-shift as a function of internuclear distance in the NH molecule.
contribute to a larger g-shift. Cross comparisons of isovalent neutral and ionic molecules show the same trend: D g H ŽNH. - D g H ŽOHq. and D g H ŽPH. - D g H ŽSHq. . Going from the neutral to the ionic molecule the nuclear charge will increase by one unit and thereby increase the spin-orbit coupling. SOC will also increase for the same nucleus when going from the neutral atom to its ion. D E is also slightly smaller for OHq than NH w15x and this energy difference may provide additional interpretation of the higher g-shifts for the ions. If the assumption of a negative two-electron gauge correction is correct, then the calculated g-shift is slightly magnified. We made a further correction of the g-shift for O 2 at the CAS1 level by adding the two-electron term from Christensen. This additional correction gives g H s 2.004820, which is in very good agreement with the experimental value, g H s 2.004838, deduced by Bowers et al. w23x. The g-shifts for NH and S 2 are plotted as a function of internuclear distance in Figs. 1 and 2. A similar performance has been obtained in the previous work on NH and O 2 molecules w7x, where the first order corrections were not accounted for. The g-shift for NH is decreasing with internuclear distance while the opposite phenomenon occurs for the O 2 w7x and S 2 molecules ŽFigs. 1,2.. The individual contributions to the g-shift of the NH molecule are plotted in Fig. 3. First order contributions are almost independent of internuclear distance. The distance dependence of the g-shift is attributed to second order terms. The most important contribution to the second order term originates from the SOC mixing between the ground 3 Sy state and the
Fig. 2. g-shift as a function of internuclear distance in the S 2 molecule.
M. Engstrom ¨ et al.r Chemical Physics 237 (1998) 149–158
155
Fig. 3. Individual contributions to g-shift as a function of interatomic distance in the NH molecule. The solid line shows the second order spin-orbitrorbital-Zeeman contribution. The dashed lines represents first order one-electron gauge correction Ž-P. and relativistic mass contribution Ž – ., respectively.
first excited 3 P state Žthe 3 P g state in the S 2 case.. The SOC integral between these states and the orbital angular momentum matrix elements are not strong functions of the internuclear distance for the O 2 and S 2 3 molecules. The energy difference between the ground 3 Sy g state and the first excited P g states is therefore the crucial value. The energy of these two states are plotted as a function of internuclear distance in Figs. 4 and 5. These plots together with Eqs. Ž10. and Ž11. explain the opposite performance of the NH and S 2 g-factors. In S 2 both states refer to the same dissociation limit with nonzero SOC constant. In NH the ground 3 Sy state and the first excited 3 P state refer to different dissociation limits Ž 2 P and 4 S states of nitrogen atom, respectively. with a large energy difference. The excitation energy at r s 2.6 au has slowly decreased to 3.69 eV from 3.82 at the equilibrium distance. The angular momentum is almost constant in the interval 1.6 to 2.5 au ŽFig. 6.. The spin-orbit coupling with the 13 P state is on the other hand decreasing in this interval ŽFig. 7.. No other states Ž n G 2. contribute much to the g-factor since they have small orbital angular momentum matrix elements with the ground state and large energy denominators. Spin-orbit coupling with the lowest 3 P state is thereby the most important effect describing the reduction of the g-shift with increasing intermolecular distance in the NH molecule. A similar performance of the g-factor and spin-rotational coupling constant can be predicted for the isoelectronic PH, OHq and SHq radicals.
Fig. 4. The energy of the ground 3 Sy state and the first excited 3 P state as a function of internuclear distance in the NH molecule.
156
M. Engstrom ¨ et al.r Chemical Physics 237 (1998) 149–158
Fig. 5. The energy of the ground 3 Sy state and the first excited 3 P state as a function of internuclear distance in the S 2 molecule.
Fig. 6. Angular momentum integrals. The integrals are calculated for the six lowest excited 3 P states as a function of interatomic distance in ˚ the NH molecule. Equilibrium distance is 1.9 A.
Fig. 7. Spin-orbit coupling integrals. The integrals are calculated for the six lowest excited 3 P states as a function of interatomic distance in ˚ NH molecule. Equilibrium distance is 1.9 A.
M. Engstrom ¨ et al.r Chemical Physics 237 (1998) 149–158
157
Table 4 Comparisons with experiments. Rotational Be and Spin rotational g 0 constants of diatomic molecules with the ground triplet 3 Sy state Molecule
Be Žcmy1 .
g 0 Ž10y3 cmy1 . Experiment
O2 SO S2 NH NF NCl PH PF OHq SHq a
1.445 0.721 0.295 16.702 1.205 0.647 8.536 0.567 16.802 9.136
a
y8.42536 y5.6153 y6.653 y54.66 y4.8 y7.15 y73.8 q1.8 y147.8 y165
This work, CAS1
This work, CAS2
y7.67 y6.01 y6.77 y48.0 y4.3 y5.4 y76.7 q3.1 y133.7 y158.7
y8.12 y5.86 y6.63 y48.0 y4.6 y5.3 y68.9 q0.7 y135.3 y143.3
Huber and Herzberg w15x.
All calculated g-shifts are positive except for PF, for which the correction is negative. The negative sign of the g-factor of PF is in agreement with experiment Žsee Table 4.. One aim of this study was to estimate if there was a significant difference between the results obtained with restricted Hartree-Fock Žwith only one configuration state function. at one hand and the correlated many-configuration ŽCAS1 and CAS2. results on the other. One finds that MCSCF wave functions are preferred since the disparity between ORHF and correlated methods can amount to a quarter of the total g-shift. On the other hand the difference between CAS1 and CAS2 is smaller. Regarding the increase in computational effort by enlarging the active space from CAS1 to CAS2, one can recommend an active space of moderate size. Experimental and calculated values for spin-rotational g 0 and rotational constants Be are tabulated in Table 4. The calculations show a reasonable agreement with experiment. In comparing calculated and experimental data there are two facts to be reflected upon. First, the Curl relation is an approximation to the exact spin rotational constant, and the influence of this approximation upon our results is somewhat hard to estimate. Second, a deviation in the first decimal of the g 0 value can be expected, due to the fact that the two-electron gauge correction is not included. Using the two-electron correction for O 2 and SO yields g 0 s y7.23024 = 10y3 cmy1 and g 0 s y5.7082 = 10y3 cmy1 , respectively. That is, the experimental agreement will be improved for SO but slightly deteriorated for O 2 . The calculated internuclear distance dependence of the g-factor leads to the same type of dependence for the spin-rotational coupling constant through Curl’s formula Ž10.. It means that the absolute value of the ge constant will increase with the internuclear distance and with the vibrational quantum number for O 2 , SO and S 2 but will decrease for the NH molecule. These trends are well reproduced by experimental findings w15x. The spin-rotational coupling constant for the first excited vibrational quantum state Ž Õ s 1. in NH is determined to be g 1 s y51.7 = 10y3 cmy1. The absolute value is about 6 percent smaller than the ground state Ž Õ s 0. value, g 0 s y54.66 = 10y3 cmy1. In S 2 , g 1 s y6.716 = 10y3 cmy1 , i.e an absolute value higher than for the ground state, g 0 s y6.653 = 10y3 cmy1 . The same trend is seen for O 2 and SO Žg 1 s y8.4468 and y5.72 = 10y3 cmy1 respectively. w15x. Accounting for the approximate nature of Curl’s formula we can say that agreement with experimental data for the electronic spin-rotational coupling constants is good. All qualitative trends are well reproduced. The most puzzling feature is a positive sign of the spin-rotational coupling constant for the PF radical. We have reproduced this sign though we have not obtained a simple explanation of this feature in terms of individual contributions. Analysis of contributions from the ten lowest excited states does not reveal the reason for this anomaly, and apparently some higher excited states are responsible for a delicate balance between positive and negative contributions to the spin-rotational coupling constant in the PF radical. This observation points at the problem with truncations of the sum-over state value, since such a truncation can break the delicate balance
M. Engstrom ¨ et al.r Chemical Physics 237 (1998) 149–158
158
between different contributing states. An account of electronic correlation can also be important for this balance; we note the experimental value for PF is just in the middle between the CAS1 and CAS2 results.
5. Conclusions In the present work we have explored the applicability of a linear response method for the calculations of electronic g-factors and spin-rotational coupling constants. We have used a series of diatomic molecules with a triplet ground state and with similar electronic structures for this purpose. Overall, we find results in good agreement with experimental data, concerning both the trends and the magnitudes of the total values. In particular, the anomalous sign and the magnitude for one member, PF, in the series could be reproduced. Correlated wave functions are essential for accurate calculations of the second order contributions to the g-shift, while the first order corrections are less sensitive. In some cases the calculations indicated the need to include the complete summation of the excited-state contributions, since upper excited states may provide a delicate balance in sign and magnitude of their different contributions.
Acknowledgements This work was supported by the INTAS grant 94-4089 and by the Swedish Foundation for Strategic Research. Computer time has been supported by NSC, the Swedish national supercomputer centre in Linkoping. ¨
References w1x w2x w3x w4x w5x w6x w7x w8x w9x w10x w11x w12x w13x w14x w15x w16x w17x w18x w19x
w20x w21x w22x w23x
J.E. Harriman, Theoretical foundations of electron spin resonance, Academic Press, New York, 1978. B.F. Minaev, J. Mol. Struct. ŽTheochem. 183 Ž1989. 207. M. Ishii, K. Morihashi, O. Kikuchi, J. Mol. Struct. ŽTheochem. 187 Ž1989. 209. H.G. Schreckenbach, Relativity and magnetic properties. A density functional study, Ph.D. thesis, University of Calgary, Calgary, Alberta, Canada, 1996. E. van Lenthe, P.E.S. Wormer, A. van der Avoird, J. Chem. Phys. 107 Ž1997. 2488. G.H. Lushington, F. Grein, J. Chem. Phys. 106 Ž1997. 3292. ˚ O. Vahtras, B. Minaev, H. Agren, Chem. Phys. Letters 281 Ž1997. 186. M.H.W. Gruebele, R.P. Muller, R.J. Saykally, J. Chem. Phys. 84 Ž1986. 2489. ¨ K. Kawaguchi, C. Yamada, E. Hirota, J. Chem. Phys. 71 Ž1979. 3338. L. Tomuta, M. Mizushima, C.J. Howard, K.M. Evenson, Phys. Rev. 12 Ž1975. 974. F.D. Wayne, H.E. Radford, Mol. Phys. 32 Ž1976. 1407. P. Bundgen, G. Lushington, F. Grein, Int. J. Quant. Chem. Symp. 29 Ž1995. 283. ¨ M. Tinkham, M.W.P. Strandberg, Phys. Rev. 97 Ž1955. 951. F.D. Wayne, E.A. Colbourn, Mol. Phys. 34 Ž1977. 1141. K.P. Huber, G. Herzberg, Molecular spectra and molecular structure. Constants of diatomic molecules, Van Nostrand Reinhold Company, New York, 1979. G.H. Lushington, P. Bundgen, F. Grein, Int. J. Quant. Chem. 55 Ž1995. 377. ¨ R.F. Curl, Mol. Phys. 9 Ž1965. 565. W. Weltner Jr., Magnetic Atoms and Molecules, Dover, New York, 1987. ˚ T. Helgaker, H.J.Aa. Jensen, P. Jørgensen, J. Olsen, K. Ruud, H. Agren, T. Andersen, K.L. Bak, V. Bakken, O. Christiansen, P. Dahle, E.K. Dalskov, T. Enevoldsen, B. Fernandez, H. Heiberg, H. Hettema, D. Jonsson, S. Kirpekar, R. Kobayashi, H. Koch, K.V. Mikkelsen, P. Norman, M.J. Packer, T. Saue, P.R. Taylor, O. Vahtras, DALTON, a second-order MCSCF molecular property program. ˚ Malmquist, B.O. Roos, Theor. Chim. Acta 77 Ž1990. 291. P-O. Widmark, P.A. H. Christensen, L. Veseth, J. Mol. Spect. 72 Ž1978. 438. R. Klotz, Ch.M. Marian, S.D. Peyerimhoff, Chem. Phys. 89 Ž1984. 223. K.D. Bowers, R.A. Kamper, C.D. Lustig, Proc. Roy. Soc. London. A 251 Ž1959. 565.