A density functional theory of the Fermi contact contribution to the nuclear spin-spin coupling constant

Abstract The authors' magnetic field density functional theory is extended to include electron spin-dependent interactions. Coupling the new theory with traditional spin density functional theory in the local limit yields a linear differential equation for the net spin density. The coefficients in the equation are functions of the electron density in the absence of a nuclear spin.

A little while ago we proved that the ground state only now being obtained [6], useful spin density energy of an inhomogeneous many-electron system functionals appear to exist [7][8][9][10].
in the presence of a vector potential, A(r), is a The proof of our theorem is identical to that universal functional of the electron density and the developed before.The many-electron Hamiltonian in magnetic field, B(r) [1,2].Through the use of time the presence of an inhomogeneous magnetic field, reversal arguments and the variational principle, we B(r), is found that second-order properties have a particularly simple form.In particular, the orbital portion of H = H 0 + BfB(r) • s(r) d3r, (1) the nuclear spin-spin coupling, the chemical shielding tensor, and the magnetic susceptibility tensor, where H o contains the nonmagnetic portions of the may be related to one universal functional of the Hamiltonian.s(r) is the spin density operator, and density in the absence of the magnetic field./x B is the Bohr magneton.Although the orbital con-Although our theorem as it stands allows us to tribution to the Hamiltonian, and therefore the enobtain the orbital portion of the nuclear spin-spin ergy, may readily be included, we neglect it.The coupling tensor, the electron spin terms were ex-neglect is justified because we are interested in cluded.The Fermi contact term is generally the most second-order properties where spin and orbital efimportant contribution to the coupling constant [3][4][5], fects are additive.We now fix the magnetic field just hence we must rectify our omission.In this note we like we fix the electron-electron interaction in ordishow that the contact coupling is a universal func-nary density functional theory.The Hohenberg-Kohn tional of the electron density in the absence of the theorem states that the ground state energy is a perturbing nuclear spins.We also point out that universal functional of the density.The proof is unlike the orbital case where useful functionals are identical to that used before [1].The magnetic field This completes the proof.Thus, were the universal functional, F( P0, r, r'), known, one could compute the contact contribution to J in any system from -I-Z a,' ~r7 i d3r d3r'' (2) knowledge of the zero-field ground-state density.Our previous work [2] identifies another similar where we have separated out the contributions to the functional from which the orbital contribution may energy due to the external field, v(r), and the classi-be calculated.cal electron-electron interaction.B denotes the ex- We now turn to the second point of this work: a plicit field dependence of the functional.The density method by which Eq. ( 3) may be used to obtain J also depends upon the field, without explicit knowledge of F. This can be done Of interest to us are the responses to weak fields, by taking advantage of recent developments in spin Symmetry under time reversal requires that the en-density functional theory [7][8][9][10].That theory states, ergy and density be even functions of the magnetic assuming that there is only one direction for the field for a nondegenerate ground state.Hence, the magnetic field, that the ground state energy is a lowest-order correction in the presence of a magnetic universal functional of the spin density deviation, field to the zero-field energy, E 0, and density, P0, is ~(r), where •(B2).The energy depends on the magnetic field explicitly and also implicitly through the density, sO(r) = p~ (r)-p+ (r) (6) hence the correction of t~'(B 2) to E 0 has two terms.
The coefficient of B 2 in the first is just (dZE/dB 2) and the total density, p(r).Thus, in the weak field (B = 0).The coefficient of B 2 in the second is p(2), limit Eq. ( 3) is replaced by the first correction to the density, multiplied by (~E/~p)(p= Po), the first variation of the energy E ~2) = f~(r)F'( po, r, r')~(r') d3r d3r '. (7) with respect to the density evaluated at the equilibrium zero-field density.However the variational prin-This form has a real advantage over the form given ciples requires this term to be equal to zero.There-by Eq. (3).There is a ~:(r) variational principle.fore only the explicit dependence on B of E[ p, B] That is, sO(r) satisfies contributes to the first correction to E 0. Hence we may write the second-order energy which we denote ~E(2) -o. (8) as E (2), as ~:(r) Hence, as is well known, we may determine ~(r) for one nuclear spin, say "a", and determine the energy (3) and coupling at the other spin, b, through use of the Hellman-Feynman theorem [3,4].That is, we have, where the functional, if( P0, r, r'), depends only on the zeroth-order density.To calculate the Fermi con-]'£a ]'tbJab ]J'b Ca( 4 tact contribution to the coupling we take the mag-where ~:a(Rb) is the spin deviation at R b due to a netic field in Eq. ( 3) to be, nuclear spin at R a.
Although we may proceed in general, let us con-4 7r~3(r_ R~,)/x,~, (4) sider the very recent developments in spin density B(r) = E a = a,b functional theory.These developments take the form where R,~ and /x,~ are the position and magnetic of exchange-correlation energy functionals, as well moment operator respectively of nuclear spin a.

+ Ir-r'l
Here the nuclear spin is at R.
-i'-#/,Bf~(r) ~3(r-ga) d3r d3r ' In conclusion, we have generalized our magnetic ' field density functional theory to include the Fermi (10) contact contribution to the nuclear spin-spin coupling constant.We have also related our theory to where T0, Ex, and E c are the kinetic energy, ex-traditional spin density functional theory.Combining change, and kinetic energy corrected correlation spin the variational principle with time reversal argudensity functionals, respectively, merits, we have shown that in the local limit the net We have already included the fact that only P0 spin density, ~(r), satisfies a linear partial differenappears to second order.Now we assume that both tial equation.The coefficients of this equation are the kinetic energy and exchange-correlation func-functions of the unperturbed density.For all density tionals are local functions of Po + ~: and IV( P0 + functionals currently in use, the functions may be )].We now expand the energy to second order in written explicitly.An example is given in the Ap-~.The most general form of the energy function is, pendix.