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Produced by the NASA Center for Aerospace Information (CASI) https://ntrs.nasa.gov/search.jsp?R=19710016498 2020-03-11T20:08:24+00:00Z ejy 29mmmbNhh6^ S41 ANNUAL PROGRESS REkRT NUMBER 71 JULY 15, 1970 SOLID-STA`rE AND MOLECULAR THEORY GROUP MASSACHUSETTS INSTITUTE OF TECHNOLOGY CAMBRIDGE, MASSACHUSET i S 9N-T .. 5 97 4 o nIA^ESSION NU. .o Q (PA / 35/ R AD NUMBER) Q (NASA CR OR TMX OR w 1 't I g CNJ F^Zl t7n?.^^^' A 1 4f ULA R 'E OF GP-7677 and GP-15428 22-009-483 - C -0054 ., TL4T-ANNTTAT. PRn(-RFgq REPORT , 1970 PERSONNEL i. Vii= ^y Faculty aid Senic, r Staf;" Professor George F. Ko.3ter Professor John C. Slater Professor Marlan O. Scully Professor Peter D. DeCicco Professor Robert Gilmore Professor Keith H. Johnson Professor H. Eugene Stanley Staff Members Research Assistants and Graduate Students H. Brandi T. Brown Y. Cho C. Gordon F. Harbus J. Ht rzfled J. Janse.: D. Ka ro H. S. Kim D. Klingspon R. Krasnow D. Lambeth M. H. Lee K. Matsuno S. Milosevic G. Paul F. C. Smith, Jr. Visiting Scientist J. W. Connolly Secretary Susan ,Leonard I k; r' s" I . I.' ii. TABLE OF CONTENTS Survey 1. CF Mclecular-Orbital Studies of the Sulfate Ion By The Scattered-Wave Model F. C. Smith, Jr. and K. H. Johnson 2. Computational Aspects of the A PW Method A Peter D. DeCicco 21. A Non-Spherical Modification to the Augmented Plane Wave Method II. Erratum and Addendum John W. D. Connolly 27. Invariant Operators I. Casimir Invariants Robert Gilmore 31 Mailing List 44. SEMI-ANNUAL PROGRESS REPORT No. 73 1. of Solid State and Molecular Theory Group, M. I. T. Including work sponsored by N. S. F. Grant NSF GP - 7677 and GP-15428 • A. P. R. A. SD-90 N. A. S. A. Grant NGR 22- 009-48 3 A. F. O. S. R. F44620-69-C-0054 SURVEY In the first contribution to this report, Smith and Johnion continue their application of the scattered wave method of computing molecular wave functions to the SO 2 ion. This represents the first time this method, which avoids the calcul- ation of multicenter integrals has been carried to self consistency. Professor DeCicco in his contribution discusses computational aspects of the APW method and in particular proposes the interpolation of the APW determinant as means of finding surfaces of constant energy and densities of states. In the remaining two contributions, John Connolly gives corrections and additions to his article in the last report and Professor Gilmore continues his dis- cussions of properties of the continuous groups. As the last item in this report you should find a form to be returned to en- albe us to update our mailing list. This seems an appropriate time to revise our list since the character of the investigations reported has widened. Professor Scullyts work on laser physics and quantum optics, and Professor Stanley's work on statisti- cal physics and phase transitions have already appeared in some of our earlier re- ports and will be appearing in future reports. I i nr nr 1. , SCF MOLECULAR-ORBITAL STUDIES OF THE SULFATE ION BY THE SCATTERED-WAVE MODEL F. C. Smith, Jr. and K. H. Johnson Tnfrnrh irnfit-in In previous publications and reports 112, we have introduced a self-consistent- field scattered-wave (SCF-SW) model for the electronic structures of polyatomic mole- cules. In the present report, we discuss the results of molecular-orbital calculations for the dinegative sulfate ion (SO 2 ) on the basis of this approach. A determination of the molecular orbitals of a polyatomic cluster such as SO is complicated, first of all, by its ionicity, by the presence of several "inner" electron shells, and, when conventional SCF-LCAO methods of quantum chemistry are used, the necessity of having to compute many multicenter integrals or equivalent Hartree-Fock matrix elements. Furthermore, the sulfate complex is not stable in free space, i.e., the gaseous phase, but typically exists as the anion in an ionic crystal, (e.g., K LSO4). Ex- perimental information on the sulfate ion can be extracted from the results of measure- ments on such crystals (e.g., 4 , ESR and chemical shift data). Thus in order to calcui ^.te sulfate molecular orbitals which are potentially relevant to observed chemical and phys- ical properties, it is essential for one to include, explicitly or implicitly, the stabilizing effects of the crystal environment. Ab i.nitio Hartree-Fock SCF-LCAO molecular-orbital calculations of the sul- fate ion have recently been published 5 . Also available in the literature are the results approximate SCF-LCAO 6 , SCCEI-LCAO ^, CNDO 8 , and 9of PP Wolf (WH) type calculations on SO4 Because of the differences among such LCAO methods, e.g., the choices of atomic orbital basis functions and the approximations to multicenter integrals, the results of these calculations are not in close agreement with each other. This incon- sistency is evident in the comparison of LCAO orbital energies and electronic charge dis- I , aSCF MOLECULAR-ORBITAL. STUDIES O TW THE SUL.FATE ION 3. tributions of SOS shown in Tables VIII and IX, respectively. The ab initio SCF-LCAO work is reported to have required a total of 7. 5 hours of computational time on a large-scale computer, most of it involved in the evaluation of multicenter integrals over the component atomic orbitals (a small Gaussian basis set including sulfur 3d orbitals). There are no multicenter integrals in the SCF-SV' ap- roach. As we shall demonstrate, stable molecular orbitals can be calculated for SOS by the latter technique with relatively little computational effort to few minutes c a n an IBM 360 Mod. 65 computer), including sulfur-s, p, d, oxygen-s, p, and higher partial- wave components in the SW bi sis set. It will be shown that the SCF-SW results are sim- ilar to these calculated with the ab initio SCF-LCAO method with respect to the ordering of the occupied valence and core orbitals and with respect to the magnitudes of the core- orbital energies. An analysis of the electronic charge distribution in the sulfate ion on thy; basis cf the SCF-SW model is also described in this report, followed by a discussion of the dependence cf the results on the choices of local exchange potential and stabilizing charge. The SCF-SW Model 1,2 Consider the sulfate ion in its equilibrium tetrahedral configuration, as illus- trated schematically in Fig. 1. In the SCF-SW model, we partition the space of the mol- ecule into three contiguous regions: I. Intraatomic: the region within nonoverlapning spheres centered on the con- stituent sulfur and oxygen atoms. II. Interatomic: the region between the inner atomic spheres and an outer sphere (the "Watson sphere") centered on the sulfur atom and surrounding the entire cluster. III. Outeratomic: the region outside the Watson sphere. These artificial spherical boundaries are rr ,st generally assumed to be touching. The atomic spheres are pictured as being relatively small in Fig. 1 for the purpose of illus- tration only. The actual sphere radii depend on the nature of the model Hartree-Fock potential chosen to initiate the SCF-SW calculation.yAt an arbitrary point r of the molecule, the initial model potential is repre- sefited as a superposition _ 4 V( r ) = VS (Ir-R 0 p+E VO (I r -IR (1)j=1 of spherically symmetric Hartree-Fock-Slater free-atom and f-ee-ion potentials centered at R j (j= 0, ... 4 ; R0 = center of cluster), including the Slater Xa 10 statistical approxi- mation V Xa (r) _ -6a [(3/8n ) p (r) ] (L) o t 4. SCF MOLECULAR-ORBITAL STUDIES OF THE SULFATh: ION to exchange correlat i on. The latter depends only c.n the local electronic charge density r(P) and on the choice of exchange coefficient a. Particular values of a = 1 and . = 2/3 have. been tested (see below). The superposition (1) is then spherically averaged within each atomic sphere j and in the outeratomic region (III). In region (II), the starting potential is that bas--L or a constant charge density equal to the ^verage value of the true charge density over the interatomic volume. To stabilize the orbitals of SO L we have added a contribution to the potential4 11 similar to that originally adopted by Watson for the calculation of free-ion Hartree• Fock wavefunctions. We distribute a positive charge uniformly around the outer spherical boundary separating the interatomic and outeratomic regions. This "Watson sphere" is therefore a crude appro.^imation to the stabilizing effects of a crystal environment. Charges of +2 (as indicated in Fig. 1) and +1 have both been tested. In Tables I through III we have listed (in Rydbergs and at every fourth radial mesh point) the respective intraatomic and outeratomic potentials used to start the SCF- SW calculation. These p otentials are based en a choice of Xa parameter a= 1 and stabil- izing charge of +2. Included for comparison in the same Tables are the final SCF poten- tials. The sulfur and oxygen spheres are assumed to be touching at respective radii where the corresponding starting potentials are equal. The Watson sphere is then assumed to be tangent to the oxygen spheres. The equilibrium S-O bond length in SOS is 1. 44 A 12 The sphere radii based on this bond length and on the starting potentials in Tables I through III are listed in Table IV, along with the initial and SCF average interatomic potentials. It is important to note that discontinuities of the potential across the boundaries separa- ting the intea- , and inter-, and outeratomic regions do not imply discontinuities in the molecular-orbital wavefunctions. Continuity of the wavefunctions and their first deriva- tives is ensured through the scattered-wave formalism (described below). The partitioning of the spv.ce of the molecule into bounded regions cf spheri- cally averaged potential allows one to introduce a rapidly convergent, composite partial- wave representation cf the molecular orbitals. Within each atomic sphere j of radius b. (sulfur: j= 0; oxygen: j= 1,2,3,41, we expand the orbital wavefunctions in the single-center form 14 (r) = E CLR p(E; r) Y L(r) (U< r 5 bj)L where L = ex;.(1, m) is the partial-wave (angular-momentum) index:; , C ,1 are partial-wave coefficients (to be determined) and Y L (r) are real spherical harmonics. The functions R I(E;r) are solutions of the radial Schrodinger equation rZ dr r2 dr + 1(1+2r1) + Vi ( r) - EJ R p ( E: r) = 0 (4) (3) SCF MOLECULAR-ORBITAL STUDIES OF THE SULFATE ION 5. for the spherical average Vi (r) of the superposition (1) with r • eF. nect to the j th atomic site (e.g., the potentials in Tables I through III). The radial functions must be finite at the origin r = 0 of each atomic sphere. the solutions are generated by outward numerical integration of Eq. (4), using the Noumerov technique 13 , for each trial energy parameter E and partial-wave component 1• In the outeratomir region the orbitals are expanded with respect to the center of the cluster in the representation III (r) = E Do Rout (F; r)Y l (r 1 (bW s r r ^)L where b is the radius of the Watson sphere. The functions R out (E; r) are solutions of a radial Schrodinger equation similar to ;4) for the spherical average of the potential in the outeratomir region. For localized molecular orbitals, these radial functions must decay exponentially at large distances from the molecule. The solutions are generated by inward Nournerov integration of the radial equation for trial values of E and 1. For the interatomic region we expand the molecular orbitals in the multicenter partial-wave representation ll(r) 130Lj1(K (r-A 0 1) YL(r-R O ) ( b0 s 1r- NO s b W; ) 4 + E E A jL f1 (K 17-- -A j 1) Y IJ(r-A i ) (b j < IrA j ^)j = 0 L or-fl0l < bW ) in which K e (E-VO ) i and h (l)1 (Kr) (Ek 1a in the ordinary APW basis, and leads to additional terms in the secular equation matrix elements. This complication we believe to be unnecessary for the following reasons; (1) For satisfactory convergence of the energy eigenvalues, even in a trans- ition element solid, it is not necessary to include terms up to I = 12. If one takes the summation up to .Q= 5 (corresponding to convergence of <.005 Ry. ), then the number of :oupled equations reduces to 36. For a non-transition element this reduction would be even greater. (2) Symmetry will reduce the number of coupled equations further. For ex- ample, in a cubic solid, the first non-zero V LM (cf. Equation (9) of Ref. 1) is for L=4. NON-SPHERICAI. MCDIFICATICN TC THE AUGMENTED PLANE WAVE METHOD 29. If we make the reasonabel assumption that the 1' im are unperturbed from their spheri- cal values for 1>2, then the only coupling is among the 1= 2 functions, and this can be removed by using cubic harmonics as has been noted3. (3) Since the nor.-spherical perturbations are expected to have a small effect on%the radial junctions, we can use an iterative technique which essentially decouples the equations. If P hn ( " ) represents the radial function after the n th iteration, then we can solve the inhomogeneous equations: + 1 "( P )) = (n+1)lm (n g (r) P + E,lm pm C m v# in : 1' m' Y 1' rn' (n) where gim (r)=1 W 1) -E+VO (r)r2 If the n = 0 terms are set equal to the spherical solutions, then this set of equations should converge quickly. The Numerov integration method, currently used in most ver- sions of the APW program, can also be used on this type of inhomogeneous equation, thereby taking advantage of the speed of this method and also minimizing the pain of re- programming. The Numerov method is u: dally applied to equations of the form; P" (r) = g ( r ) P(r) but it can also be used on equations of the form P" (r) = g(r) P(r) +f(r) In this latter case, the recursion relations for P becomes P n+1 = (1 -h 2 gn+1 /12)-1[(2+5h2gn/6)Pn-(1-h2gn-1 /12)Pn-1 + h2(fn+1+ l Ofn +fn-1 ) /12] where only the last three terms have to be added in modifying the existing programs. 30 NON-SPHYRI( Al. MODIFICATION TC THE ALTGMI ,,'N ,r •:i) PLANY WAVI •' MYTHOD I lie fe rences I John W. D. Cunnolly, M. I. T. SSMTG Semi-Annual Progress Report No. 72, 26 (1970). 2. The author is grateful to Professor P. DcCicco for pointing out this omission. 3. L. Kleinman and R. Shurtleff, Phys. Rev. 188, 1111 (196,)). INVARIANT OPERATORS I. CASIMIR INVARIANTS Robert Gilmore I. Introduction There has been a great deal of interest in the invariant operators associated with a Lie group. These operators may be used to classify states belonging to an irreducible representation s , to approximate spectra in atomic +. systems L , and as models for Hamiltonians in nuclear systems ] and systems possessing phase transitions Although the general properties of these invar- iant operators have been well-kncwn for a long time s ' 4 and some have been constructed explicitly3, 5, 6, they have not yet all been explicitly constructed for the simple classical Lie groups. Because of their widespread use, we feel that an explicit constructive determination of these operators is approp- riate. la the following sections, we construct the Casimir invariants for the unitary (A n y), smplectic (C n), and orthogonal (B .. , D 1) groups. We also construct (some of) the invariants for non-semi-simple groups which may be constructed from semi-simple groups by Inonu-Wigner contraction. 31. I^ r32. INVkRIANT CPFRATORS I. CASIMIR INVARIANTS H. Construction of Invariants The Casimir invariants of a Lie algebra are functions of the gener- ators which commute with all the generators. In short, they form a basis for the Lie algebras universal enveloping algebra. Their general properties are well known 1,4 : I. They are symmetric homogeneous polynomial functions of the generators which are invariant under the automorphisms induced by the Weyl group of re- flections. L. The product of the ord rs of the n functionally independent and irreducible invariants of a simple rank n algebra is equal to the order of the Weyl group of reflections for that algebra. 3. On a compact algebra, the eigenvalue of the invariant c°^ on an irreducible representation with highest weight M h is &i(Mh) = fi (Mh +R) - f (R) (1) Here f i consists of those terms in the polynomial invariant which contain only the mutually commuting generators Hi of the Cartan subalgebra, and R is de- fined as usual- R = Z E (position roots) (L) fi (H) are called the associated invariants. 4. The Jacobian J (H) = det a II is cor-ipletely antisymmetric under the JWeyl group. We will make an explicit constructive determination of the invari- ants using the proceedure outlined in steps 1 1 -4 1 below, then use observations 1-4 above to prove that the operators so constructed are in fact irreducible polynomial invariants. INVARIANT (`PERATORS I. CASIMIR INVARIANTS I t . faithful matrix representation of the generators Xa is determined: X a faithful M a ( 3 )^ > 2 1 . The secular equation for the L,ie algebra element a s Xs , in this represent- ation, is determined det 11 aa Ma - a I ^I =E ^` f (aa ) = 0 (4) 3 1 . The symmetrized functions f (Ma ), with the matrices Ma replacing arguments aa , obey [ fi (Ma), MR ] = 0 (5) 4 1 . The general polynomial invariants are then f ( a), since they obey a commutation relation isomorphic to (5) [ f i (Xa ) , X ] = 0 (51) III. Invariants of the Unitary Groups U(n) (An-1) The generators of ti,! group U(n) are U Hi i ^ iU j Eel-ej j [Ut j, Ur s ] = UL s 6 j - Ur j 6i s ( 3) A faithful n x n matrix representation of these generators is given by UL -» M^ : (M^ ) = d, d. (4 )j j j rs tr is Zhe faithful matrix representative of the Lie algebra element a is alj UlI i I^ a j 11 (5) 13. i , 34. INVARIANT OPERATORS I. CASIMIR INVARIANTS ' The determinant of this matrix is given by a it a 12 . a in jlJ2..-in e... e det (I al II _ 11 t 2 ... in j1 2 Jn (6) n! i The symbols e ' ' e ... are the usual Levi-Civita skew tensors jl j2 jlj2 ... jn e = in (7) The secular equation of the matrix (5) is computed using the sub- =z stitution a a ij - 6ij (8) The delta function serves to contract indices. The resulting secular equation is det ^) aij _ I ^^ _ E ( -X)n-rfr(aiJ) (9) ` e. • • ail . ,, air ei i... jr I^r+1 "' In f (a i ) _ L 1 • i t 1r+1... In jr r j r! ( n-r)! ( 91) The corresponding 1 -iants (neglecting an unimportant numerical factor) for the unitary groups U (n) , are C U(n) (UL) E: t I 1 U i I U i r ej 1 ... j Ir+1... In 1 r r+1 "' n i 1 it (10) r IV Invariants of the Symplectic Groups US p (2n) : (en) is The group USp (2n) is a subgroup of U(2n). This suggests the possib- ility that the invariants of USp(2n) might be I subduced , from those of U(2n). The t INVARIANT OPERATORS I. CASIMIR INVARIANTS g4nerators Z j= ±1, +2,... fn) obey Z Ir l (r I = H (1) z j sign (i) sign (j) Z j-i (2) [Z Ij , Z rs ) = sign ( jr) {Z s b j+ Z-fir bs+ Z1 r 6 j +Z s bT- J (3) p SO(2n+1) (4) 35. Since i i j i , one of the other values of i is equal to j i • Without loss of generality, we can choose i 2 = j l . Then the schematic contraction may be rearranged: I s; r 1 r f , Proceeding by induction, we conclude 1) C r vanishes if r is odd 2) The n invariants C r SO(Zn+l) of SO(2n+I) (r 1, 2,. n) are SO(2n+l)C r (Xiji I I xi i XZr-I, 2r1*** 2r Zr+l — 2n+1 1 2 x e if -•j2r 12r+1 '2n+1 X. x 2 J2r-l'j2r (6) 36. INVARIANT OPERATORS I. CASIMIR INVARIANTS S0(2n+0 (5) J4 u<3 37 `1 I INVARIANT OPERATORS I. CASIMIR INVARIANTS i2 e So(2 n+ 1) '2 12r- i2r 12r ^2r.1 12►. • • • • 1, i, 12 8 8 i2 • • ^2r-' 12r. j 2r 12r (61) .-I VI Invariants of the Orthogonal Groups SO(2n) (Dn) t a The arguments of Sec. V for the algebras B n may be repeated essent- ially verbatim for the algebras Dn . The only salient point is that C_nS0(tn)(XLJ) is a perfect square, since there are no lines left to contract the skew tensors. The invariants C SO(2n)r (Xis ), r= 1, 2, ...n-1 are given by (V. 6), replacing l2n+1 by fZn. The invariant C n0(Zn)(XL ) isJ CSO(2n) (X ) _ eiliZ...i2n-1 LZn X. .. . X.n LJ 1112 t2n-1 1 2n. (1 ) (11) 38• INVARIANT OPERATORS I. CASIMIR INVARIANTS Tr pp SO(2 n) 1On (Xi d VII Irreducibility and Functional Independence We observe that the operators giver by expressions [IIi. 10, IV. 4, V. 6, VI. 1 1) are symmetric homogeneous polynomials in the group generators 2) which commute with all the group generators 3) and are left invariant under automorphisms induced by the Weyl group. 4) The product of the orders of these inva riants is in each case equal to the order of the appropriate Weyl group. . 5) The number of invariants is equal to the rank of the algebra. SU(n) ='A n-1 has n-1 invariants, since C1U(n) (Ul ) = E . H. = 0 (1 ) i=1 L rr INVARIANT OPERATORS I. CASIMIR INVARIANTS 39. These invariant operators fulfill observations (1) and (2), section II. We also observe, from [Ill. 1 and III. 101 , that the associated in- variants * for the unitary groups are given by k n-1 (H) = F 17 Hi k= 1,2,...n (2) i r# i s r=1 r The associated invariants for the algebras Cn , Bn , D can easily be determined from [ IV. 1, IV. 4, V. 1, V. 6 and VI. 1 ] r C ,B ,D kf n n n(H)=F ri H 2 (kJn for D)k ir# is r=1 i t n f D n (R) = n nH n t=1 i From (2) and (3) the value of the Jacobian J (H) = det 11 OC +I on any representation with highest weight R may be determined: J(Mh+R) _dim Mh (4) J(O+I7) The invariants [III. 10, IV. 4, V. 6. Vi. 1 ] or alternatively (2) and (3) therefore also satisfy observation (4), Sec. II. Therefore these invariants are the func;- tionally independent irreducible polynomial invariants of the classical Lie groups. The spectrum of these invariants on the compact real forms of these al- gebras has been determined using observation 3 of Sec. II. VIII Invariants of Subgroups -A Caution It is often possible to determine the invariants of a group 0' if it is embedded in a larger group 39 whose invariants are known. Thus we were able to subduce the invariants of Cn , B and D using canonical embeddings. This technique may be used to suggest the structure of (subduced invariants), but these * See above, Sec. II, observation 3. (3) 40. INVARIANT OPERATORS I. CASIMIR INVARIANTS operators may not belong to the center of the subgroup's universal enveloping algebra at all. In each case, the subduced operators must be shown to be in- variant operators on the subgroups. For example, SO(3) has generators J 3 ► J+: (13P i±1 _ }Jf J+, J, ] = 2J3 (1 ) C1 O(3) = J32+ i(.J+J- + J- J+) (2) The generators J 3 , J+ form a subalgebra of the original algebra. The ' invar- iant of this algebra subduced from: the well known Casimir invariant (2) is C'1 (J 3 , J+ ) ' J3 2 ( 3) This is easily seen not to be an invariant of the subalgebra, for IC^1 (J3' J+ )' J+ ] = J3 J+ + J } J3 (4) u In f,--,.et, this particular algebra has no invariants at all. IX Invariants of Contracted Groups t It is possible to form new group structures by the process of contracir tion8'9' 10 If the group 4' is a contraction of the group, then operators C (X') may be subduced from the invariant operators C (X) onll^yl. These op- erators C'1 (X') may be or may not be invariant operators on If the con- traction process used is the one proposed by Inonu and Wigner 8 , then opera- tors subduced from invariants will ren.iain invariant operators. Example 1: The invariants of the groiup SO(4) are C1S0(4)(X) _ E Xi.t i

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