260 ELECTRIC FIELD IN HCN AN ANALYSIS OF THE GRADIENT OF THE ELECTRIC FIELD IN HCN BY ANDRE BASSOMPIERRE Laboratoire d’Electronique et de Radioklectricite Received 31st January, 1955 Numerous data from nuclear quadrupole resonance experiments and Hertzian spectro- scopical techniques are now available on nuclear quadrupole couplings. The chief difficulty in obtaining the quadrupole moments lies in our ignorance of the electrical gradients acting on the nuclear quadrupole in molecules and solids. We have 1 tried here to solve this problem in a case which is attractive in its relative simplicity, the molecule HCN. Evaluation of the gradient of the electric field, to which the nucleus of nitrogen is submitted, requires a rather accurate knowledge of the electronic structure. Ordinary valence schemes are not sufficient for this purpose.For instance, they are not able to provide information on the perturbations of inner atomic shells. Such perturbations, which have small effects on the usual properties of molecules, cannot be neglected here, since they destroy the spherical symmetry of these shells. This is especially important for nitrogen, since the 2s atomic electrons give a lone pair coupling with the quadrupolar moment of the nucleus which is quite considerable. The method of self-consistent field 2 extended to molecules by Roothaan, provides a convenient way for obtaining a suitable electronic wave function. This method being now well known, we shall give only a brief account of it. The molecule HCN is linear and the intemuclear distances 3 are H-C = 1*064& C-N = 1*156& The axis Oz is directed from H to N.A N D R ~ BASSOMPIERRE 2 6 1 We consider an antisymmetric normalized wave function for the fourteen elect- rons of the molecule of the form where the #i are an orthonormal system of space functions and a, p the two functions of electronic spin.P is an operator which permutes the electrons, designated by upper indices, and p its parity. We suppose that the functions +f are developed in an arbitrary subspace X, described by functions X,, +i = xqcqi- 4 The function $ is now determined only by the parameters Cgi. The best values of these parameters may be obtained by searching for those for which the electronic energy is a minimum. For this purpose, we use the complete Hamiltonkin in- cluding all electronic interactions ; we neglect only the spin orbit coupling, which gives a fine structure.It is now easy to show that the unknown parameters C g j , written as column matrices Ci, verify the equation S-lFCi = EiCj, where F and S are the following matrices : The ej are scalars whose values (with change of sign) give the energies of vertical ionization, that is to say, before disturbances of nuclei and other electrons of the molecule occurs. For solving eqn. (1) we must first choose a sub-space X. If it were complete, the equation would be equivalent to Fock's equation. We have taken as a basis eleven real atomic functions : ( 1 s ) C ~ (1s)N7 (1s)H7 (2s>C7 (2pz)C7 (2s)N, (2pZ)N, (2pX)G (2pX)N, (2py)C7 (2p,)N The indices designate the corresponding nuclei.We shall take the functions xq in the order written above. We have used the atomic functions of Duncanson and Coulson.4 These functions are orthonormal for each atom, and so the base x is not orthogonal on the whole. The atomic functions are used only as a vectorial basis which permits one to introduce easily the singularities in space constituted by the nuclei and also the correct symmetry conditions around these singularities. The introduction of these functions into the equation allows the evaluation of many integrals. In spite of all the studies which have been made by many authors, their evaluation remains tedious, especially because the molecule is heteronuclear. We have tried to calculate each of them as carefully as possible; the approximation introduced in these calculations is probably the main limitation to the precision of our wave function.When these integrations have been performed, it is possible to solve the equation. It has the form of an ordinary eigenvector equation of the eleventh order, and moreover, its operator S-1F is a quadratic function of some of the eigenvectors Ci (we count those corresponding to effectively filled levels only). Such an equation may be solved by iteration. Taking a set of vectors Ci we form262 ELECTRIC FIELD I N HCN the operator S-1Fand then solve the eigenvector equation, and so on, until the field becomes self-consistent. The order of the equation may be reduced by symmetry arguments, since the functions s, p z and px, py belong to different irreducible representations of the group C,, of the molecule.In this way we get one equation of seventh order and two of second order. In addition, a simple perturbation calculation shows that 1s states are not appreciably mixed with the other states. Finally, the cquation is reduced to a set of equations whose orders are respectively 1, 1, 5, 2, 2. The levels which are filled are only those corresponding to the fin-st-order equations, the three lowest levels of the fifth order equation and also the lowest one of each of the two last equations. We shall give to these levels respectively the numbers 1, 2, 3, 4, 5, 8, 10 (according to the preceding notations these are the second indices in the parameters Cpi). We have solved the fifth order eigenvectors equations using the excellent method of M a y ~ t .~ At the tenth iteration the field becomes self-consistent. The con- vergence is alternate ; probably it would be possible to accelerate the rate of con- vergence by taking mean values of the results of two preceding iterations. We obtained C33 = 0.165, c34 = 0.002, c35 =- 0.159, c88 = c10, 10 = 0.575, C43 = 0.439, c44 = 0.001, c45 = 1.200, c98 == c11, 10 == 0.657, C53 =- 0.388, C54 = 0.267, C55 = 0,189, C11 = C22 = 1. c63 =- 0.680, c@ = 0.016, c65 =- 0.037 C73 = 0.324, C74 =: 1.042 C75 = 0.484, (the other parameters are zero). Kusch, Hustrulid and Tate 6 have found experimentally that the energies for vertical ionization of levels 8 and 5 are equal to 13-7 and 26-3 eV. We have obtained for these levels 17.3 and 26-4eV.The evaluation of the first of these levels is apparently less satisfactory than the second one mainly because of a differentiation effect. In fact, the eighth iteration gave for the level 8 an energy of 14.1 eV in better agreement with the experimental result 13.7. On the other hand the corresponding parameters Cgg and c98, given by the eighth iteration, were very close to those of the last iteration (c88 = clo, 10 = 0.569, cgg = c11,10 = 0.662). So we may consider that the agreement with experiment is good. With the preceding wave function it is now possible to calculate the gradient of the electric field acting on the nucleus of nitrogen. The molecule having a cylindrical symmetry around the axis Oz, we need only to calculate the mean value of b2V/3z2. The part qe of this due to the electrons may be written as the spherical co-ordinates being centred at the N nucleus.For evaluating each of these integrals, we have developed the function X, and X4 in spherical harmonics around the N nucleus. This may be achieved with the help of some of CouBson's formulae 7 using Bessel functions of half order and imaginary argument. We have calculated the radial integrals by numerical integration. Seeking a high precision we were obliged to integrate rather far from the nucleus, more than 1 A ; this proves that, if the main part of the quadrupolar coupling comes from electrons surrounding the nucleus, all the electrons of the molecule must, in fact, be considered. At a distance from the nucleus of the order of h/mc, our wave function is no longer valid due to the relativistic perturbation, which introduces a spin-orbit coupling.It is difficult to obtain information on the electronic structure inside this sphere; nevertheless we think that the relative error is rather small. We have still to consider the gradient of the electric field due to the nuclei H and C. For its evaluation, we have taken account of the vibrations of the molecule.ANDRE BASSOMPIERRE 263 A study of the normal modes of vibration 8 shows that at ordinary temperature we need only to take an average value over zero-point rotations of the H atom. It was possible to neglect the influence of these vibrations on electrons, since the electrons being mainly concentrated on the CN group are not appreciably per- turbed by small rotations of the H atom.Finally we obtain the gradient q of the total electric field equal to q = - (2-729)e (the first Bohr radius being the unit of length). Simmons, Anderson and Gordy 9 have found experimentally that the nuclear quadrupole coupling eqQ is -4.58 Mc/s. From this, we may infer that the nuclear quadrupolar moment of N14 is Q = (0.0071) x 10-24cm2. It is quite difficult to state the accuracy of this result. We hope to check it by calculations on other molecules. We intend to try and extend the preceding results to give an approximate understanding of the electronic structure of solids such as ICN and BrCN. 1 Bassompierre, J. Chim. Phys., 1954,51,614 ; Compt. rend., 1954,239,1298 ; 1955,17. 2 Roothaan, Rev. Mod. Physics, 1950, 23, 69. 3 Simmon, Anderson and Gordy, Physic. Rev., 1950 77, 77. Nethercot, Klein and Townes, Physic. Rev., 1952, 86, 798L. 4Duncanson and Coulson, Proc. Roy. SOC. Edin., 1944, 62, 37. 5 Mayot, Ann. Astrophys., 1950, 13,282. 6 Kusch, Hustrulid, and Tate, Physic. Rev., 1937, 52, 840. 7 Coulson, Proc. Cumb. Phil. Soc., 1942, 38, 210. 8 Penney and Sutherland, Proc. Roy. SOC. A, 1936, 156, 654. 9 Simmons, Anderson and Gordy, Physic. Rev., 1950,77, 77.