INTERPRETATION OF DATA OBTAINED FROM MÖSSBAUER, ESR, SUSCEPTIBILITY, OPTICAL AND XPS MEASUREMENTS
1980; EDP Sciences; Volume: 41; Issue: C1 Linguagem: Inglês
10.1051/jphyscol
ISSN2777-3418
Autores Tópico(s)Molecular spectroscopy and chirality
ResumoMolecular o r b i t a l c a l c u l a t i o n s a r e designed t o c a l c u l a t e t h e e lect r o n i c and magnetic s t r u c t u r e f o r molecules o r c l u s t e r s . With t h e ca l cu l a t ed e l e c t r o n i c s t r u c t u r e (eigenvalues & and e i g e n v e c t o r s y o f t h e r e l e v a n t e l e c t r o n i c Hamiltonian) it i s i n p r i n c i p l e pos s ib l e t o eva lua t e spec t roscopic d a t a from corresponding expec ta t ion va lues , where Ti rep re sen t s t h e i n i t i a l and ?gf t h e f i n a l e l e c t r o n i c s t a t e , and 1s t h e appro r i a t e ope ra to r (i .e. d(P) f o r de r iv ing e l e c t r o n and s p i n d e n s i t i e s , (31q -T$+/T' f o r e l e c t r i c f i e l d g r a d i e n t s , $ f o r d i p o l e moments and t r a n s i t i o n s , ~ + 2 g f o r magnetic momments) . Inc luding energy spacing and Boltzmann populat ion of e l e c t r o n i c s t a t e s i n t h e c a l c u l a t i o n l ead t o temperature dependent da t a . Experimental d a t a ob t a i ned from Mijssbauer, ESR, s u s c e p t i b i l i t y , o p t i c a l and XPS measurements a r e r e l a t e d t o t h e expec ta t ion va lues mentioned above. The c a l c u l a t i o n a l procedure w i l l be descr ibed , and comparison of c a l c u l a t e d and measured d a t a w i l l be presented. 1. Introduct ion.The mutual i n f luence of motion is j u s t a s it would be i f t h e nue l e c t r o n i c s t r u c t u r e c a l c u l a t i o n s and c l e i were a t r e s t a t t h e p o s i t i o n they ocspec t roscopic measurements he lps t h e theocupy a t t h a t same i n s t a n t (Born-Oppenheir e t i c i a n t o t e s t t h e r e l i a b i l i t y of t h e mer approximation) . Then t h e e l e c t r o n i c va r ious approximations u sua l ly involved i n system i s descr ibed by t h e Hamiltonian ( i n h i s c a l c u l a t i o n s and t h e expe r imen ta l i s t a .u.) t o i n t e r p r e t h i s da t a . Molecular o r b i t a l A n N 4 H = ~ ( f v ? ; & , ) + ~ 1 (MO) c a l c u a l t i o n s a r e designed t o ca lcu lar.4 ;>j 1; 51 (1) t e t h e e l e c t r o n i c and magnetic s t r u c t u r e of molecules o r c l u s t e r s . I n t h e p re sen t communication t h e b a s i c p r i n c i p l e of t h e c a l c u l a t i o n a l procedure ( s e c t i o n 2 ) i s described, some common approximate methods a r e mentioned ( s e c t i o n 3 ) , molecular expec t a t i on va lues a r e def ined ( s e c t i o n 4 ) , and c a l c u l a t e d e l e c t r i c f i e l d g rad i en t s ( s e c t i o n 5 ) , e l e c t r o n charge d e n s i t i e s a t t h e nucleus ( s ec t ion 6 ) , i n t e r n a l magnetic f i e l d con t r ibu t ions ( s e c t i o n 71, magnetic s u s c e p t i b i l i t y and gfac tors ( s e c t i o n 81, ene rg i e s ( s e c t i o n 9 ) , d ipo le moments and XPS i n t e n s i t i e s ( s e c t i o n l o ) a r e presented f o r some compounds and compared wi th exper imenta l da t a . 2 . Basic P r inc ip l e . A molecular system cons i s t i ng of N nuc le i and n e l e c t r o n s i s descr ibed by t h e t o t a l Hamiltonian, which inc ludes motion and i n t e r a c t i o n of n u c l e i and e l e c t r o n s . However, we can r e s t r i c t ourse lves t o t h e e l e c t r o n i c p a r t of H t o t a 3 t h i s assumes t h a t t h e e l e c t r o n s a d j u s t themselves t o new nuc lea r p o s i t i o n s s o r a p i d l y , tha t a t any i n s t a n t t h e e l e c t r o n a where Z k and Rk a r e t h e charge and t h e pos i t i o n vec to r of nucleus k , and gi i s t h e p o s i t i o n vec to r of e l e c t r o n i. Each of t h e n e l e c t r o n s may be a s soc i a t ed with a oneN e l e c t r o n func t ion yi, c a l l e d s p i n o r b i t a l , which is a product of a spa_tial and a s p i n p a r t : Gi(? ,$ ) = %(;I X;(S>. The t o t a l n-electron wavefunction i s now b u i l t up a s an antisymmetrized product of molecular s p i n o r b i t a l s Such an a r r a y of s p i n o r b i t a l s , known a s S l a t e r determinant , i s t h e s imp le s t wavefunc t ion which s a t i s f i e s t h e antisymmetry p r i n c i p l e . The e l e c t r o n i c p a r t of t h e tot a l energy of t h e system i s der ived from eqs. ( 1 ) and ( 2 ) by Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1980116 C1-96 JOURNAL DE PHYSIQUE According to the variational principle,the best molecular orbitals are obtained by varying the one-electron functions yi until E achieves its minimum value.The condition for the minimum of energy leads to the differential equations, also termed HartreeFock (HF) equations where Ei are orbital energies, and where A J. is the Coulomb operator and K is the I j exchange operator. The orbitals yi obtained from the solution of these equations (4) are referred to as HF-MO's. For molecular systems the direct solution of eq. (4) is impracticable. It is more convenient to approximate the HF-MO's by a linear combination of atomic basis orbitals qP (LCAO) where the (m are now optimized so as to give minimum energy. This procedure leads to a set of equations, which are algebraic (Roothan equations) l and for any molecular size much easier to handle than the HF equations which are differential equations : with the Fock matrix the overlap matrix between AO's ( P, and $, the bond order or density matrix the one-electron energy matrix and the four-center Coulomb and exchange integrals J( P V , A6 ) and K( P A , v b ) . The Roothan equations (6) can be transformed to a standard eigenvalue problem,which yields MO energies E i as well as LCAO-MO coefficients c Ai. Since the Fock matrix depends on the LCAO-MO coefficients c i , c5i through PAC , the solutions of (6) can achieve selfconsistency by stepwiseimprovement of c ~ ; : with an initial guess of cai the Fock matrix Fp, is evaluated and the Roothan equations are solved; this leads to a new set of coefficients ca; , and so on. 3. Approximations.Eq. (6) describes closed-shell cases. A generalization of Roothan's equations for open-shell cases Is given by separate representation of exchange L interaction for or and R spins in eq. (6) . Calculations in which all integrals are worked out explicitly using Roothan's procedure are termed ab initio calculations. The amount of computation time requiredfor such accurate calculations, especially for large systems, can became extremely high. Thus, for many chemical and physical problems, where qualitative or semiquantitative knowledge of the MO's is sufficient to extract the necessary information, approximations to the Roothan's equations may be considered. A number of approximate MO theories have been developed, as for example Hfickel methods3-' (nonextended , extended, extended and iterative), zero differential overlap methods6-' (CNDO,INDO,NDDO,NINDO) , pseudo-potential methods lo-' (one-elec.tron approximation of two-electron contributions by model potentials such as in the X, method). In the present contribution the electronic structure and expectation values for molecules and clusters have been derived on the basis of the iterative extended Huckel theory (IEHT), in some cases including a limited configuration interaction calculation' ; and for some smaller molecules a model potential method12 was used. 4. Expectation Values.In order to compare experimental data with calculated values we define the expectation value of a spatial one-electron operator 8 [= d' ($1 for electron charge and spin density, ( 3pq-r2&) / r5 for electric field gradient, 2 for dipole manent) 4 0 ) = 5, n; Ilyie(;) yi(;) d ; , (11) where ni is t h e number of e l e c t r o n s i n MO Yi. S u b s t i t u t i n g t h e LCAO s o l u t i o n (5) int o (11) y i e l d s From eq. (12) it is obvious t h a t a l l molec u l a r expec ta t ion va lues c o n s i s t of a molec u l a r p a r t Ppu , which i s d i r e c t l y der ived from t h e M O ' s Yi ( s e e eq. ( 5 ) and ( 9 ) ) , and of an atomic p a r t t o > a t , which can be eva lua ted from t h e b a s i s A O ' s at , depending on (i) whether both +, and qy belong t o sit e k ( d i r e c t o r valence con t r ibu t ion ) (ii) whether only 9, blongs t o s i t e k ,but +, t o a l igand o n ) , and (iii) whether both, qp and t & belong t o l i gands ( l a t t i c e o r l i gand c o n t r i b u t i o n ) . 5 . E l e c t r i c F i e l d Gradient and Quadrupole S p l i t t i n g . A s a f i r s t example f o r ca lcu lat i n g expec ta t ion va lues according t o eq. (12) we d e r i v e e l e c t r i c f i e l d g rad i en t s (E FG). The corresponding t enso r ope ra to r f o r a l l n e l e c t r o n s and N n u c l e i i n t h e molec u l e i s I n ca se t h a t co re e l e c t r o n s of t h e Msssbaue r i so tope a r e excluded from t h e LCAO basis s e t i n o rde r t o save computer time ( i . e . 2 6 f o r 5 7 ~ e : I s ... 3p ) , eq. (13) has t o be modified by t h e app rop r i a t e Sternheimer s h i e l d i n g func t ion y ( r ) Case (i) and (iii) of s e c t i o n 4 reduces t o t h e well-known express ions
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