TRANSFORMATION OF REAL SPHERICAL HARMONICS UNDER ROTATIONS

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1 Vo. 39 (008) ACTA PHYSICA POLONICA B No 8 TRANSFORMATION OF REAL SPHERICAL HARMONICS UNDER ROTATIONS Zbigniew Romanowski Interdiscipinary Centre for Materias Modeing Pawińskiego 5a, Warsaw, Poand Stanisław Krukowski Institute of High Pressure Physics of the Poish Academy of Sciences Sokołowska 9/37, Warsaw, Poand Abraham F. Jabout Department of Chemistry, The University of Arizona, Tucson, AZ USA (Received February 8, 008) The agorithm rotating the rea spherica harmonics is presented. The convenient and ready to use formuae for = 0, 1,, 3 are isted. The rotation in R 3 space is determined by the rotation axis and the rotation ange; the Euer anges are not used. The proposed agorithm consists of three steps. (i) Express the rea spherica harmonics as the inear combination of canonica poynomias. (ii) Rotate the canonica poynomias. (iii) Express the rotated canonica poynomias as the inear combination of rea spherica harmonics. Since the three step procedure can be treated as a superposition of rotations, the searched rotation matrix for rea spherica harmonics is a product of three matrices. The expicit formuae of matrix eements are given for = 0, 1,, 3, what corresponds to s, p, d, f atomic orbitas. PACS numbers: Ar, Fx, Mb, Dx 1. Introduction In the previous paper [1] the agorithm rotating the compex spherica harmonics was presented. The proposed agorithm has the foowing properties: It does not use the Euer anges [ 4]. Therotationis determinedby therotationaxis and onerotation ange [5]. (1985)

2 1986 Z. Romanowski, St. Krukowski, A.F. Jabout For fixed anguar momentum number,, the rotation matrix, D, which rotates the compex spherica harmonic, is a product of three matrices. The main idea of the agorithm is to spit the rotation into three steps: 1. Express the compex spherica harmonics as the inear combination of canonica poynomias, matrix B.. Rotate the canonica poynomias, matrix C. 3. Express the rotated canonica poynomias as the inear combination of compex spherica harmonics, matrix A. It was shown in Ref. [1], that if B is invertibe, then B = A 1, and the searched rotation matrix D is a product: D = A 1 C A. (1) From the above, it foows that the matrix C depends ony on the canonica poynomias. In the present paper, the three steps procedure, for rea spherica harmonics is appied. Let us denote (for fixed ) the rotation matrix for rea spherica harmonics by D r. Further, et denote by Ar the matrix expressing the rotated canonica poynomias as the inear combination of rea spherica harmonics. Since the matrix C depends ony on the canonica poynomias, then: D r = (Ar ) 1 C A r. () Since the matrix C was derived in Ref. [1] for = 0,1,,3, then the ony thing required to define the rotation matrices for rea spherica harmonics is the definition of matrix A r. In the present paper the matrix Ar is derived for = 0,1,,3, what corresponds to s,p,d,f atomic orbitas.. Definitions According to the Condon Shortey phase conventions [4,1], the compex spherica harmonics are defined for m as: Y m (θ,ϕ) = N m P m (cos(θ))e imϕ, (3) where N m is a normaization factor: N m = i m+ m [ + 1 4π ] ( m )! 1/ (4) ( + m )!

3 Transformation of Rea Spherica Harmonics under Rotations 1987 and P m (v) is an associated Legendre function defined by Rodrigues formua [6, Eq..11] vaid for v 1 and m 0: P m (v) = (1 v ) m/! d m+ dv m+ (v 1). (5) Rea spherica harmonic, Y m (θ,ϕ), is defined [7] as a inear combination of compex spherica harmonics: 1 [Y m (θ,ϕ) + ( 1) m Y m (θ,ϕ)] for m > 0, Y m (θ,ϕ) = Y m (θ,ϕ) for m = 0, (6) i [Y m(θ,ϕ) ( 1)m Y m (θ,ϕ)] for m < 0. In Ref. [1] it was shown, that r Y m (θ,ϕ) for m 0 is a compex poynomia of x,y,z: r Y m (θ,ϕ) r Y m (x,y,z) = N m (x + iy)m ( m)/ k=0 γ (m),k rk z k m, (7) where (x,y,z) is a point in Cartesian coordinate system defined by the point (r,θ,ϕ) in the spherica coordinate system, hence, reations hod: r = x + y + z, z = r cos(θ), y = r sin(θ)sin(ϕ) and x = r sin(θ)cos(ϕ). In Eq. (7), the coefficient γ (m),k is defined as )( k γ (m),k = ( 1)k ( k ) ( k)! ( k m)! and ( m)/ is the argest integer number ess than ( m)/. Mutipying Eq. (6) by r and substituting Eq. (7) we obtain the rea poynomia of x,y,z: r Y m (θ,ϕ) r Y m (x,y,z) 1 [Y m (x,y,z) + ( 1) m Y m (x,y,z)] for m > 0, = r Y m (x,y,z) for m = 0, i [Y m (x,y,z) ( 1) m Y m (x,y,z)] for m < 0. The function Y m (x,y,z) is a Cartesian representation of the rea spherica harmonic. For exampe, for = 1 we get: Y 1 1 (x,y,z) = 3/(4π)y/r, Y 0 1 (x,y,z) = 3/(π)z/r, Y 1 1 (x,y,z) = 3/(4π)x/r. (8) (9)

4 1988 Z. Romanowski, St. Krukowski, A.F. Jabout 3. Rotation of rea spherica harmonics in R 3 Let us introduce the sphere of any radius and the center ocated at the origin of coordinate system, and denote by (θ,ϕ) and (θ,ϕ ) two points ocated on it. It was proved [ 4], that for compex spherica harmonics the reation hods: Y m (θ,ϕ) = M= d () m,m Y M (θ,ϕ ). (10) This reation means, that for fixed, the set of functions S = {Y m (θ,ϕ)} for m =,..., is compete. By definition (6), rea spherica harmonic, Y m (θ,ϕ) is a inear combination of two compex spherica harmonics Y m (θ, ϕ) and Y m (θ,ϕ) with the same anguar momentum number. Thus, because of competeness of set S, we have Y m (θ,ϕ) = M= d () m,m YM (θ,ϕ ), (11) () where d m,m denotes the eement of (searched) rotation matrix Dr. As was indicated in Section 1, the ony required thing to define the matrix D r = (Ar ) 1 C A r is to find the matrix Ar (). The eements ã k,m of matrix A r are defined by the equation [1]: Q k (x,y,z) = r m= ã () k,m Ym (x,y,z). (1) In this equation, the canonic poynomia, Q k (x,y,z) for k =,...,, is represented as a inear combination of rea spherica harmonics. Due to the specific seection of Q k () (x,y,z), the eements ã k,m can be easiy found from the poynomia representation of Y m (x,y,z), defined in Eq. (9). Since the (0) rea spherica harmonic for = 0 is constant, then d 0,0 = 1. The resuts obtained for = 1,, 3 are presented in the foowing subsections Expansion coefficient for = 1 Let reca the canonica poynomias Q k 1 (x,y,z), for k = 1,0,1 Q 1 1 (x,y,z) = x, (13a) Q 0 1(x,y,z) = y, (13b) Q 1 1(x,y,z) = z. (13c)

5 Transformation of Rea Spherica Harmonics under Rotations 1989 Then, based on Eqs. (7), (9) and (1) the matrix A r 1 has the form: A r 1 = 4π 3 The matrix A r 1 is invertibe / 0. (14) 3.. Expansion coefficient for = Let reca the canonica poynomias Q k (x,y,z), for k =, 1,0,1,: Q (x,y,z) = yz, (15a) (x,y,z) = xz, (15b) Q 1 Q 0 (x,y,z) = xy, Q 1 (x,y,z) = x y, Q (x,y,z) = z x y = 3z r. Then, based on Eqs. (7), (9) and (1), the matrix A r has the form: A r = 4π (15c) (15d) (15e). (16) The matrix A r is invertibe Expansion coefficient for = 3 Let reca the genera set of canonica poynomias Q k 3 (x,y,z), for k = 3,...,3 Q 3 3 (x,y,z) = x(4z x y ), (17a) Q 3 (x,y,z) = y(4z x y ), (17b) Q 1 3 (x,y,z) = z(z 3x 3y ), (17c) Q 0 3(x,y,z) = xyz, (17d) Q 1 3(x,y,z) = y(3x y ), (17e) Q 3(x,y,z) = x(x 3y ), (17f) Q 3 3(x,y,z) = z(x y ). (17g)

6 1990 Z. Romanowski, St. Krukowski, A.F. Jabout Appying Eqs. (7), (9) and (1) it can be verified that the matrix A r 3 has the form: π A r 3 = / / (18) 3/ / / 10 0 The matrix A r 3 is invertibe. 4. Possibe appication In Density Function Theory (DFT) [8,9] the fundamenta equation is the Kohn Sham eigenprobem Ĥ KS ψ µ = ε µ ψ µ. (19) For moecuar systems, this equation is often soved by Linear Combination of Atomic Orbitas (LCAO), with ψ µ (r) = j c µ,jχ j (r), where χ j (r) are so caed basis functions and c µ,j are the expansion coefficients. The LCAO method transforms the Kohn Sham functiona eigenprobem, Eq. (19), to the agebraic generaized eigenprobem Hc = εs, where the eements of matrices H and S are given by h j,k = j(r)ĥksχ k (r)dr, s j,k = j(r)χ k (r)dr, (0) R 3 χ R 3 χ where denotes conjugate compex. The main cost of LCAO method is: Evauation of integras h j,k and s j,k. Soution of generaized agebraic eigenprobem Hc = εs. Often, the basis function χ j (r) is represented as a product of the radia part and the spherica part: χ j (r) χ j (r,θ,ϕ) = R j (r)u j (θ,ϕ). When spherica part is a compex spherica harmonic, i.e. U j (θ,ϕ) Y m (θ,ϕ), then matrices H and S are compex. When spherica part is a rea spherica harmonic, i.e. U j (θ,ϕ) Y m (θ,ϕ), then matrices H and S are rea. Since the soution cost of generaized eigenprobem is four times higher for the compex matrices than for the rea matrices [10,11], it is desirabe to appy the rea spherica harmonics to save the computationa time.

7 Transformation of Rea Spherica Harmonics under Rotations 1991 The evauation of the matrix eements h j,k is a very compicated task. Generay, the integras h j,k can be cassified as one-, two-, three- and fourcenter integras. The evauation of these integras can be substantiay simpified, when the rotations and transations of basis functions χ j (r) are used. Since the radia part of χ j (r) does not change under rotations, then to rotate χ j (r) ony the spherica part Y m (θ,ϕ) must be rotated. When the spherica part of χ j (r) is represented by a rea spherica harmonic, then the agorithm described in the present paper might be usefu. 5. Summary The rotation of the rea spherica harmonic was anayzed. The rotation was defined by the rotation axis and the rotation ange. The rea spherica harmonic defined in the fixed coordinate system was expanded as a inear combination of the rea spherica harmonic in the rotated coordinate system. The present manuscript heaviy depends on the previous paper, where the rotation of the compex spherica harmonics was considered. For both cases, the rotation matrix is determined by two matrices. Since one matrix C is the same for compex and rea spherica harmonics, the ony difference is in the matrices A and A r, which were easiy obtained. The present agorithm might be usefu in computationa quantum chemistry. REFERENCES [1] Z. Romanowski, S. Krukowski, J. Phys. A40, (007). [] M.E. Edmonds, Anguar Momentum in Quantum Mechanics, Princeton University Press, New York [3] D.A. Varshaovitch, A.N. Moskaev, V.K. Khersonskii, Quantum Theory of Anguar Momentum Techniques, Word Scientific, Singapore [4] E.O. Steinborn, K. Ruedenberg, Adv. Quant. Chem. 7, 1 (1973). [5] J.D. Foey, A. van Dam, S.K. Feiner, J.F. Hughes, R.L. Phiips, Introduction to Computer Graphics, Addison Wesey Professiona, New York [6] M. Abramowitz, I.A. Stegun, Handbook of Mathematica Functions with Formuas, Graphs, and Mathematica Tabes, Dover Pubications, New York 197, cbm/aands/toc.htm [7] E.O. Steinborn, Adv. Quant. Chem. 7, 83 (1973). [8] R.G. Parr, W. Yang, Density-Functiona Theory of Atoms and Moecues, Oxford University Press, Oxford [9] W. Koch, M.C. Hothausen, A Chemist s Guide to Density Functiona Theory, Ed. Wiey, New York 000. [10] J. Stoer, R. Buirsch, Introduction to Numerica Anaysis, Springer, NY 004. [11] G.H. Goub, C.F. Van Loan, Matrix Computation, The Johns Hopkins University Press, Batimore [1] E.U. Condon, G.H. Shortey, The Theory of Atomic Spectra, Ed. Cambridge University Press, UK Cambridge 1970.