Vox. UMz 25, NUMsaR 2S 21 DZCaMazR 1970 0. Bely, J. Phys. 8: Proc. Phys. Soc., London 1, 718 (1968); O. Bely and P. Faucher, Astron. Astrophys. 1, 37 (1969). G. W. F. Drake, to he published. C. Bchwartz, private communication. M. Lipeles, R. Novick, and Tolk, Phys. Rev. Lett. 15, 690 (1965). 9C. Artura, Tolk, and R. Novick, Astrophys. ¹ J. J. 157, L181 (1969). B. C. Elton, I . J. Palumbo, ¹ and H. R. Griem, Phys. Rev. Lett. 20, 783 (1968). A. H. Gabriel and C. Jordan, Nature 221, 947 (1969); J. %. M. Neupert and M. Swartz, Astrophys. 160, L189 (1970); A. B. C. Walker, Jr. , H. R. Ruggge, Astro Astrophys. 5, 4 (1970). R. W. Schmieder and R. Marrus, Phys. Rev. Lett. 25 1245 (1970). A. S. Pearl- Phys. Rev. Lett. 24, 703 (1970); R. S. Van Dyke, Jr., C. E. Johnson, and H. A. Shugart, Phys. Rev. Lett. 25, 1403 (1970). R. Marrus and B. %. Schmieder, Phys. Lett. 32A, 431 (1970). R. Marrus and B. %. Schmieder, preceding Letter [Phys. Rev. Lett. ~25 1689 (1970)j. Calculation of the Indirect Spin-Spin Coupling Constant in HD Molecule»' Many-Body Perturbation and T. P. Das of Utah, Salt Luke City, C. M. Dutta, N. C. Dutta, DePa&ment of Physics, University Utah 84122 (Received 15 June 1970) The linked-cluster many-body perturbation approach has been applied to the study of indirect nuclear spin-spin coupling constant JHD in HD molecule. The complete set of states used were the bound and continuum states of H2+ molecular ion with the internuclear separation appropriate to H2 molecule. Our calculated value of JHD through the Fermi con'tact interaction mechanism ls +42. 57 Hz in good agreement with the most lecent experimental value of +42.7 + 0.7 Hz. Although the indirect spin-spin interaction between nuclei in molecules has been utilized extensively for qualitative understanding of electronic structures in molecules, its quantitative calculation even for the simple molecule HD has proved to be a formidable task, with the current situation far from conclusive. ' The various theoretical approaches utilized so far can be divided broadly into two categories. In the first category is the conventional second-order perturbation as first proposed by Ramsey and Purcell. The difficulty with such an approach is that one requires a knowledge of the complete set of ground and excited states of the molecule, which is not usually available. To obviate the knowledge of excited states, variation-perturbation procedures have been used by a number of authors. ' In one class of such calculations, a diagonal-type perturbation procedure" was used, the second-order energy due to the hyperfine field of one nucleus being minimized t:o obtain the first-order perturbed wave function of the molecule. This function was then utilized to calculate the cross term in the second-order energy involving the other nucleus to obtain PHD. The difficulty with this procedure was that the second-order nuclear self-coupling ' " ' ' energy is infinite in nonrelativistic theory, no real minimum thus being attainable in a variational approach. A second class of variation perturbation calculation has attempted to extremize the cross terms in the second-order energy proportional to JHD using variational functions which describe the first-order perturbation due to both nuclei. The difficulty with this procedure is that the cross term due to two per turbations has by itself no minimum and one does in fact get oscillatory behavior as the number of parameters '' increased. In this paper, we have revived the perturbation approach in a form that: meets the major difficulty, namely, a knowledge of a complete set of states for the molecule. This is accomplished by using the linked-cluster many-body perturbation theory (LCMBPT), where a neighboring Hamiltonian X„ for which the complete set of states can be obtained exactly, is used as the starting point for a perturbation treatment of 4X =X-X . In our work here, a multiple perturbation approach is used in conjunction with the LCMBPT, using the sum of 6+ and the two hyperfine interaction Hamiltonians XH' and XD' associated with the two nuclei. Fol the speed of convergence of the perturbation approach, it is nec- " Vox. UMz 25, NUMszR 25 21 DmcaMszR 1970 essary that the basis states chosen (and hence g) describe the behavior of the charge densities near the nuclei reasonably well. In particular, a one-center choice for X might be expected to be rather inadequate for the present problem since this would require the inclusion of very high angular-momentum states to describe properly the charge densities near the nuclei. With this consideration in view, the H, molecular-ion Hamiltonian was chosen for X . This has the dual merit of providing a basis set that is both exactly de- ' rlvRble RQd has the deslx'RMe cusp behavior Rt. the nuclei. The procedul e of cRlculRtlon follows b10adly the same lines as that employed in handling atomsie systems subject to external pexturbations. Only a very brief description will be presented here with emphasis on some of the points that have special importance for the present problem. The starting Hamiltonian, corxesponding to two noninteracting electrons in the H, molecule framework, is given (in atomic units) by " ' ~ f, Xo= Z i= x , ( 1 1 rHi roi leading to AX =1/r». The net perturbation Ham1ltonlRQ K' composed of 4+ Rnd the hyperflne 1Qteraction terms is then given by X' = (1/ri2) where the +XH +XD, X„' (A = H or D) are given by representing the Bohx magneton Rnd nuclear magnetic moment of nucleus A. and I„ and S; the nuclear and electron spin operators. Following the usual linked-cluster perturbation is givapproach, the total energy corxeetion p, & Rnd p, ~ ~ en by AE = . Z &4. (-, 0)X.'(O) I .'(0, ")I+.&, (4) the suffix I. and other quantities in Eq. (4) having their usual meanings in linked-cluster perturbaThe indirect nuclear spin-spin intion theory. teraction Hamiltonian has the form " KHD =h JHDIH- ID. The spin-spin coupling constant PHD can be evaluated by equating the expectation values of both sides of Eq. (5) over the nuclear spin states with magnetic quantum numbers MH =IH and MD =ID. The expectation value ~HO of the left-hand side of Eq. (5) corresponds to the energy derived from Eq. (4), keeping one order in XH' and one order in XD' and all possible orders in 1/r». In diagrammatic representation, the corresponding diagrams must contain the X„' and 3*D' vertices once, whereas the 1/r» vertex can occur any number of times. 81Dce the spin Hamiltonlan 18 isotropic, it i.s sufficient to woxk with only the z-component term in Eq. (5). For the one-electron basis set for the diagrammatic evaluation of we have utilized the exact bound and continuum states of X which correspond to H2+ molecular-ion wave functions for the internuclear distance R =1.4 a.u. These are expressed in the form ~» ~;(&, u, V) =A;(&)M;(V)e"', where A,-(X) and M,-(p) are functions of the elliptic coordlnatesq A. = (r H+ r.n)/R~ p, = (riH D)/ R, Rnd r,-H and r&D are the distances of the jth electron from H and D nuclei. For the bound states, we have to determine both the energy eigenvalues as well as the eigenfunctions described by Eq. (6). This requires the solution of the appropx iate second-ordex differential equations for A;(X) and M;(ii) which are coupled by the energy and a separation constant A. ;. The eigenvalues and eigenfunctions for some of the lower bound states axe already available in the literature. For the higher bound states, only eigenvalues are available" and we had to solve the necessary continued-fraction equations to obtain A; and M;(p, ). The functions A;(i) were obtained by solving the corresponding differential equaFox' the continuum statesq tion numerically. the eigenvalues are, of course, e; =k'/2. Again A; and M;(p) are obtained by solving the requisite continued-fraction equations and A,.(a) through numerical integration of the coxxesponding differential equation. ' The diagrams involved in the calculation of ~HD Rx'e similar 1Q form to tho8e oDe eDcountex'8 in the perturbation of atomic systems in an external field. The external fields in the present molecule are the hyperfine fields of nuclei at the sites of the electrons. For the hyperfine operators K„' and KD' we have utilized wiggly lines terminating with dots in the diagrams. The lowest-order diagram for the present calculation is of second order involving one order each in XH' and XD'. The higher orders involve additional vertices associated with 1/r». Figure 1 shows the second-oxder and Rll of the third-order diagrams. The typical fourth-order diagrams involving two hyperfine vertices and two 1/r» ver~ ~ " " " ~ -r VOLUME PHYSICAL REVIEW LETTERS 25, NUMBER 25 i 21 DECEMBER 1970 F ($) FIG. 1. Second- and third-order diagrams for the spin-spin interaction in HD molecule. The wiggly line denotes the Fermi contact operators KH' and RD' . tices are shown in Fig. 2. We have included all the distinct diagrams that are obtained from those in Fig. 2 by taking account of time reordering and permutations of the vertices among themselves and between hole and particle lines. The most time-consuming aspect of the evaluation of diagrams was the calculation of matrix elements associated with the vertices. The hyperfine vertices only require the density of the wave funcHowtions at the nuclei and are straightforward. ever, the matrix elements of 1/r», particularly those involving continuum states, require special attention, since elliptic coordinates are involved. We have utilized the procedure developed by RMenberg" for variational molecular bound-state calculations. Our computer program for this purpose was checked by comparing our values for a few two-center two-electron integrals using Slater orbitals with currently available tables for such integrals. It should be pointed out that because of the use of elliptic coordinates, the integration over the continuum states involves a somewhat different multiplying factor than in atomic work, " " Q, - (2/vR)'J dk. The second-order diagram (1a) represents the contribution to JHD from two noninteracting electrons in the ground state of a H, molecular ion with internuclear separation of R =1.4 a.u. The third-order diagrams (1b)-(1g) represent the contribution to JHD from the first-order effect of 1/r» on diagram (1a). Of these, diagrams (lb)(1e) represent the role of the passive HartreeFock interaction between the electrons which converts the electronic wave functions from ' FIG. 2. Some typical fourth-order diagrams for the spin-spin interaction in HD molecule. ' those of H, molecular ion to the H, molecule. Diagrams (1f) and (1g) represent the influence of two-particle self-consistency effects in the perturbed state of the system when the nuclear hyperfine interaction is switched on. The contributions to JHD from various diagrams are listed in Table I. For the second-order diagram (la) we have separately listed the contributions from gerade and ungerade particle states. A rather substantial cancelation is seen to occur between the two types of excitations in this order. For the diagrams (1b)-(lg), the contributions are listed after the gerade and ungerade cancelations have been carried out. , wherever they occur. For the sake of brevity, only the net contribution from all the fourth-order diagrams is listed. The third-order result is seen to be about one-half of the second order. This somewhat slow decrease up to this order is a result of the rather severe cance1ation between gerade and ungerade excitations in second order. However, on comparing individual diagrams, even the diagram with the largest contribution in third order is found to be a factor of 4 smaller than the smaller (gerade) of the second-order diagrams. This situation is reminiscent of the hyperfine interaction problems in atomic oxywhere the lowgen, nitrogen, and phosphorus, " " 1697 PHYSICAL REVIEW LETTERS VOr. UME 25, NUMBER 25 Table I. Contributions 21 DECEMBER 1970 from various diagrams to JHD. Contributions Order of perturbation Diagrams (Hz} Second order (la) gerade (la) ungerade -385.65 458.67 73.02 Subtota1 Third order -74.76 (lb) -10.16 (1c} (ld) -47.60 (le) 47.41 24.90 24.90 -35.31 (1f} (1g} Fourth order Subtotal Subtota1 Grand total Experiment est order core-polarization diagrams have rather substantia1 eance1ations which make the higher orders of crucial importance. In fourth order, there is strong cancelation among the various diagrams due to the gerade and ungerade symmetry and spins. Additionally, there is also substantial cance1ation among different types of diagrams leading to the net relatively small contribution shown in Table I, namely, about 6.7% of the second-order result. We have carried out rough estimation of important fifth-order diagrams and find that this strong cancelation persists and that the net contribution in this order is expected to be no more than a cycle. Our result compares very favorably with the experimental value of +42.7+P.'7 Hz. The experimental value does include small contributions from second-order effects involving dipolar and orbital hyperfine interactions. Several estimates" of these contributions have been made in the literature which are consistently rather " small. It is our feeling" that these mechanisms more than the quoted range of experimental error. We would like to point out that the present procedure can also be used to evaluate these additional contributions to JHD. It is worth mentioning that in contrast to LCMBPT calculations on atomic systems, laddering procedures to sum over selected classes of diagrams are not suitable in the present approach. This is because the diagrams (ld) and (le) which are the progenies of the usual holehole and hole-particle ladders are comparable in importance with the other diagrams in third order hand it is not appropriate to selectively apply laddering procedure to any one class of diagrams. do not contribute 4.86 42.57 42.7 +0.7 Instead, the correct procedure for the present problem is to sum all diagrams for each order and such a sum is seen from our work to show satisfactory convergence. It should be pointed out that in ea, rlier work, Matcha. and Brown" and Goodisman" had used a zero-order H, + basis set, as in the present work, and utilized variation-perturbation procedures to calculate the energies of the ground and excited states of H, molecule with rather satisfactory results. The convergence observed here using the same starting Hamiltonian for a complex property such as the spin-spin coupling constant demonstrates the general validity of the ionic basis set as a starting point for perturbation calculations on diatomic molecules. It has been possible to carry out this calculation of the indirect spin-spin coupling constant due first to the increased sophistication attained in recent years in applying LCMBPT procedures to atomic systems and secondly to the advance in computing techniques which makes it practicable to work with one-electron continuum states of molecular systems. It is expected that the present procedure would be easily applicable to other hyperfine and magnetic properties of hydrogen molecule involving double perturbation by external sources, such as the rotational moment and spin- rotation constant. We are grateful to Professor D. R. Bates for supplying us a table of eigenvalues prior to publication. Thanks a.re also due to Professor R. L. Mateha for useful discussions. *Work supported by National Science Foundation. VQLUME PHYSICAL REVIEW LETTERS 25, NUMBER 25 N. F. Ramsey and E. M. Po.rcell, Phys. Rev. 85, 143 (19M). F. Ramsey, Phys. Rev. 91, 303 (1953). 3E. A. G. Armour, Chem. Phys. ~49 5445 (1968). E. Ishiguro, Phys. Rev. 111, 203 (1968). T. P. Das and R. Bersohn, Phys. Rev. 115, 897 (1959) . M. Stephen, Proc. Roy. Soc., Ser. A 243, 274 ¹ J. J. {1957}. D. E. O'Reilly, J. Chem. Phys. 36, 274 (1962). J. Schaefer and R. Yaris, J. Chem. Phys. 46, 948 {1967). S. Ray, M. Karplus, and T. P. Das, unpublished. J. Goldstone, Proc. Boy. Soc., Ser. A 239, 267 {1967). C. M. Dutta, A 1, 661 (1970). ¹ 21 DECEMSER 1970 demic, New York, 1968), Vol. 1V, p. 13. ' D. R. Bates, U. Opik, and G. Poots, Proc. Phys. Soc., London, Sect. A 66, 113 {1963). 5K. Riidenberg, Chem. Phys. 19, 1459 (1951). 8M. Kotani, A. Amemiya, E. Ishiguro, and T. Kimura„ Tables of Molecular Integrals {Maruzen Co. Ltd. , Tokyo, 1966). H. P. Kelly, Phys. Rev. 131, 684 (1963). H. P. Kelly, Phys. Rev. 173, 142 (1968}, aud 180, 66 (1969) . N. C. Dutta, C. Matsubara, B. T. Pu, and T. P. Das, Phys. Rev. 177, 33 (1968), and Phys. Rev. Lett. 21, J. 1139 (1968). T. F. Wimmett. Phys. Rev. 91, A476 (1963). Sign determined hy I. Ozier, P. N. Yi, A. Khosha, and F. Ramsey, Bull. Amer. Phys. Soc. 12, 132 (1967). S. D. Mahanti and T. P. Das, Phys. Bev. 170, 426 (1968) . 22R. I, . Matcha and W. B. Brown, Chem. Phys. 48, 74 (1968). J'. Goodisman, J'. Chem. Phys. 48, 2981 (1968). ¹ C. Dutta, and T. P. Das, Phys. Rev. D. R. Bates, K. Ledsham, and A. L. Stewart, Phil. Trans. Roy. . Soc. London, Ser. A 246, 216 {1963). D. R. Bates and B. H. G. Beid, in A.defences in Atomic and Molecular Physics, edited by D. R. Bates (Aca- J. Formation and Strncture of Electrostatic Collisionless Shocks* D. %. Forslund and C. R. Shonk Los Alamos Scientific Laboratory, University of California, Los Alamos, (Received 4 November 1970) Neav Mexico 87544 A one-dimensional, two-species numerical-simulation code has been used to study the formation and structure of coQisionless electrostatic shocks formed by two colliding plasmas and by a plasma striking a perfectly reflecting piston. Collisionless shocks are = vD/e formed in hydrogen up to a maximum piston velocity of M — s 3.5. Shocks with velocities up to M 4 have been produced which have a predominantly laminar structure accompanied by strong collisionless dissipation. - Theoretical models for electrostatic shocks in the absence of a magnetic field have been discussed by a number of authors. ' ' Lom-Machnumber [M=vo/cs&2, where cs=(T, /m;)'~'] shocks have been experimentally produced by Alikhanov, Belon, and Sagdeev' and Taylor, Baker, and Ikezi, ' and numerically simulated by Mason' and Sakanaka, Chu, and Marshall. ' In this Letter me discuss formation of stable, highMach-number electrostatic shocks by means of the particle-in-cell simulation technique treating both the ions and electrons exactly with the mass ratio of hydrogen (1836). Since the electrons are treated exactly instead of isothermally, the critical Mach number of 6 does not apply. The shocks described here are mell approximated by a rapid change from one equilibrium state to another with little or no fluctuations in the shock front (although perhaps having large fluctuations behind the front), and thus correspond most closely to the "laminar"' ~ rather I-1, ' model. Initial conditions. —The shocks discussed here are created in tmo mays. The fiI st method consists of a plasma of ions and electrons moving to the right with Mach number M, as shown in Fig. 1(a), with the plasma initially (T =0) just in contact with the right-hand, perfectly reflecting mall. The plasma is sustained at the left by continually injecting a Maxwellian plasma. The second method ls similar but with the above plasma filling only the left half of the box and an equaldensity plasma with Mach numbeI of opposite sign occupying the right half. This configuration clearly should generate tmo shocks which dlffel' only statistically. For M &1 a shock will readily form on the basis of the ion-ion instability'0 in one dimension. Shocks formed with M &1 do so by means of the nonlinear process described below. Theory of formation and structure. —In the tmo lllltlal conditions used the method of shock forthan "turbulent"