ESI-3D: Electron Sharing Indexes Program for 3D Molecular Space Partition

Transcription

1 ESI-3D: Electron Sharng Indexes Program for 3D Molecular Space Partton Insttute of Computatonal Chemstry (Grona), 006. Report bugs to Eduard Matto: or The Electron Sharng Indexes (ESI) The electron sharng ndexes (ESI) account to some extent for the electron sharng of a gven par of atoms. The ESI computed wth ths program are calculated from the exchange correlaton densty (XCD), whch reads as follows: ( ) ( ) ( ) γ xc ( x, x ) = γ ( x ) γ ( x ) γ ( x, ) () x and measures the dvergences of the par densty, ( γ ) ( x, x ), a true electron par dstrbuton, and a fcttous par densty constructed as the product of two ( ) ( ) ndependent one electron dstrbutons, γ ( x) γ ( x ). Theore, the XCD ntegrates to N, snce the densty and the par densty ntegrate to N and N(N-), respectvely: γ ( x, x ) dxdx xc = N () The XCD s an appealng functon because t fulflls several nterestng propertes. Frst of all, t s non-negatve defnte; the number of fcttous pars of electrons for a model of ndependent electrons s always greater than the actual number of pars. Secondly, by defnton, n the case of non-nteractng electrons the functon s zero. And n thrd place, ntegrated over the whole space, t gves the number of electrons n the system, N, whch enables a chemcal meanngful molecular space decomposton of the N electrons of a gven system. For a sngle determnant wave functon (Hatree-Fock, or a Kohn-Sham orbtals one) the par densty can be wrtten n terms of the frst order reduced densty matrx (- RDM): r r r r ( ) r r γ γ ( xx ) = (3) γ ( ) ( ( x ) ) x γ ( x x ) () r r () r r ( x x ) γ ( x x )

2 () r where γ ( x x r ) s the frst order reduced densty matrx (-RDM) whch for the case r r x = x reproduces the densty. Thus smplfyng the expresson of the XCD to read: r r () r r γ ( x = γ ( x x ) (4) xc, x) In 975, Bader and Stephens ponted out that the ntegraton of the XCD upon two dfferent regons of the space A and B leads to a measure of the correlatve nteractons between the electrons n these two moetes. However, t was not Bader the frst to propose such a quantty as an ESI. In 993 Fulton 3 proposed an ESI based n the framework of the QTAIM by usng the -RDM n followng manner: () r r ( ) ( ) [ / = ] ( ) r r A B x x ( x x ) [ / ] r r δ, γ ; γ ; dxdx, AB (5) whereupon the pseudo-square root of the -RDM reads () r r ( ) [ / ] / * r r γ x x = λ η ( x ) η ( x ), ; k k k k (6) λ k and η k (x) beng the natural occupances and natural spn orbtals, respectvely. It s more convenent to express Eq. (5) n followng manner: (, = δ A λ λ S ( A) S (, / / j (7) S (A) s the dagonal element of the atomc overlap matrx (AOM) n terms of natural spnorbtals ntegrated over the doman of atom A: r r r S ( A) = η ( x ) η j ( x ) dx (8) A For sngle determnant methods Eq. (7) smplfes to: (, = δ A S ( A) S (, (9) where S (A) now corresponds to the AOM n terms of spnorbtals. For monodetermnantal wave functons t concdes wth later defntons gven by Ángyan 4 or by Fradera 5 for the quantum theory of atoms n molecules (QTAIM) partton of the

3 molecular space nto atomc domans. 6 Fradera et al. 5 also proposed the localzaton ndex as: A) δ λ ( A ) = = S ( A) S ( A), (0) Chemcally nterestng can be also the atomc delocalzaton whch s smply calculated as follows: ( A ) = δ δ, B A () and some authors recognze ths quantty as the atomc valence. 7 Fnally s worth notcng that these XCD based ESI can be calculated from dfferent parttons. The frst partton attempted to calculate ths ndex was reported by Wberg 8 for a Mullken-lke decomposton. 9 Ths program has been desgned to deal wth any AOM properly formatted. In partcular we have adapted the AIMPAC format, 0 whch s also the format of FUZZY package. Theore, by usng the *.nt fles generated by these programs, one can calculate QTAIM-ESI and FUZZY-ESI respectvely. The program s prepared for the calculaton of unrestrcted and restrcted sngle determnant ESI. In addton, the program provdes the possblty of splttng the ESI accordng to orbtal contrbutons. In general the ESI can be decomposed as follows: δ ( A, = ( ) ( ) S A S B = δ () j whch may gve further nsde n the orbtal contrbuton to the chemcal bondng. 3 In general an exact orbtal decomposton s done for those cases where S (A)=0 when and j belong to dfferent sets of orbtals. Ths s the case of σ-π separaton of a planar molecule. In the cases where the decomposton s not exact t s explctly ndcated n the output. For an extensve revson of these quanttes check for example erence 4.

4 Aromatcty Indexes: FLU and PDI From these ESI the calculaton of the aromatcty ndexes PDI 5, 6 and FLU 7 s also possble wth ESI-3D. The PDI s a specfc measure of local aromatcty for sx member rngs (6-MRs), where three para-related postons exst, namely (,4), (,5), and (3,6). By defnton, the PDI of a 6-MR s gven by: (,4 ) + δ (,5) δ ( 3,6) δ + PDI = (3) 3 Whle the FLU ndex of aromatcty reads: FLU = n RING A B δ ( δ ( A) α δ δ δ, (4) where the summaton runs over all adjacent pars of atoms around the rng, n s equal to the number of atoms of the rng, δ s the ESI for the atomc par A and B from the aromatc molecule chosen as a erence, and α s a constant to ensure the rato of atomc valences s greater than one, δ ( > δ ( A) α = (5) δ ( δ ( A) The program recognzes the bonds C-C, C-N and B-N for whch t takes the erence parameters from benzene, pyrdne and borazne, whch are respectvely.4,. and 0.86, for QTAIM, and.4,.5 and.63 for FUZZY parttons. If the bond s not recognzed the program uses C-C value by default and gves a warnng message. The program needs for atomc overlap matrces accordng to the format of FUZZY or AIMPAC 0 programs; 8 both programs are also freeware. When the orbtals are decomposed nto two groups the program attempts the calculaton of PDI π and FLU π. For a recent revew on electronc aromatcty ndexes check erence 9.

5 Accuracy In order to check the accuracy of the calculaton one must check that that the sum rule s fulflled: δ ( A ) ( ), Aj + λ A = δ ( A ) + λ( A ) = N( A ) = j N (6) It s of recommendable practce to ensure the accuracy acheved for N to be at least 0-3 electrons. Snce the ndexes are calculated from the AOM n Eq. (8), t s also recommendable that the followng quantty: Error( S) = S ( A) δ (7) A never ncreases over 0-3. Addtonally AIMPAC program gves the value of the laplacan of the densty wthn a basn. For an accurate calculaton ths value s usually below 0-3. Quotaton Usage Program: E. Matto, n 'ESI-3D: Electron Sharng Indexes Program for 3D Molecular Space Partton.' Grona, IQC, 006. PDI ndex: J. Poater, X. Fradera, M. Duran, and M. Solà, Chem. Eur. J., 003, 9, 400. FLU ndex: E. Matto, M. Duran, and M. Solà, J. Chem. Phys., 005,, PDI or FLU from FUZZY-partton: E. Matto, P. Salvador, M. Duran, and M. Solà, J. Phys. Chem. A, 006, 0, 508. Downloads ESI.x fle.n Tests: test.x ESI-3D: FUZZY: AIMPAC:

6 Keywords $ATOMS. Number of atoms and fles contanng the overlap matrx for each atom. PROAIM (AIMPAC) or FUZZY format expected. $ATOMS number of atoms (n) fle_.nt fle_.nt... fle_n.nt $BASIS. The number of occuped molecular orbtals (or spnorbtals f open-shell). $BASIS number of occuped (spn)orbtals $TYPE. Type of wave functon, restrcted- HF or KS wave functon (hf) or unrestrcted one (uhf) expected. If uhf then the program reads the number of the frst beta spnorbtal. $TYPE hf or uhf (number of the frst beta spnorbtal) $RING. Needed for the calculaton of aromatc propertes. Holds the nformaton of the connectvty of the rng(s). Important: the atoms of each rng must be specfed accordng to the connectvty of each rng. $RING number of rngs (nr) number of members n the frst rng (nm)... nm (the atom number accordng to ther connectvty)... number of members n the n-th rng (nmn)... nmn $GROUPS. Needed for the orbtal decomposton of the ESI and the aromatc ndexes. Frst we must provde the number of groups, and afterwards the number assgned to each orbtal accordng to the group they belong. For σ π separaton of 5 occuped orbtals t mght read: $GROUPS $FUZZY. When the program founds such keyword t uses the erences of FUZZY- ESI for the calculaton of FLU.

7 Example Benzene s gven as an example of a molecule wth a rng, whose decomposton s attempted by dfferent symmetry of occuped orbtals (benzene.es.d6h), and afterwards by dong the sgma-p separaton (benzene.es.cs). Here below a b explanaton: INPUT: $BASIS $ATOMS benze_c.nt benze_c.nt benze_c3.nt benze_c4.nt benze_c5.nt benze_c6.nt benze_h7.nt benze_h8.nt benze_h9.nt benze_h0.nt benze_h.nt benze_h.nt $TYPE hf $RING rngs found 6 members of th rng $GROUPS Ths nput requests a sngle Slater determnant closed-shell calculaton (hf) of all ESI accordng to Eq. (9) and (0). Snce the keyword $RING s specfed, the program wll attempt the calculaton of FLU, and provdng the number of members n the rng s sx, he wll also provde the PDI measure. ESI-3D wll perform an orbtal analyss of the ESI based on the decomposton gven. The FLU π and PDI π wll be also calculated because there are two groups of orbtals, and, whch wll be supposed σ and π respectvely.

8 OUTPUT Populaton and ESI analyss: Eq. (0) Eq. () Atom N (nt) loc. deloc. N N C C C C C C H H H H H H N(nt) s the populaton gven n the nput fles whch contan the AOM (*.nt) N corresponds to the populaton of orbtals n group N corresponds to the populaton of orbtals n group Eq. (9) Par DI DI DI C -C C -C C -C C -C C -C C -C C -H C -H C -H H -H H -H TOTAL: LOCAL: DELOC: TOTAL Total number of electrons calculated by Eq (6). LOCAL Total localzaton DELOC Total delocalzaton

9 δ ( δ (A) α δ δ δ RING - 6 bonds Atom par delta -->j j--> Exc(,j) dff Flu(,j) C -C C -C C 3-C C 4-C C 5-C C -C FLU = FLU = n RING A B Flu(, j) = n RING A B δ ( δ ( A) α δ δ δ PDI = References K. Ruedenberg, Rev. Mod. Phys., 96, 34, 36. R. F. W. Bader and M. E. Stephens, J. Am. Chem. Soc., 975, 97, 739. R. L. Fulton, J. Phys. Chem., 993, 97, 756. J. G. Ángyán, M. Loos, and I. Mayer, J. Phys. Chem., 994, 98, 544. X. Fradera, M. A. Austen, and R. F. W. Bader, J. Phys. Chem. A, 999, 03, 304. R. F. W. Bader, 'Atoms n Molecules: A Quantum Theory', Clarendon, 990. I. Mayer, Chem. Phys. Lett., 983, 97, 70. K. B. Wberg, Tetrahedron, 968, 4, 083. R. S. Mullken, J. Chem. Phys., 955, 3, 833. F. W. Begler-Köng, R. F. W. Bader, and T.-H. Tang, J. Comput. Chem., 98, 3, 37. I. Mayer and P. Salvador, n 'Program 'FUZZY'. Avalable from < Grona, 003. E. Matto, J. Poater, M. Solà, M. Duran, and P. Salvador, J. Phys. Chem. A, 005, 09, M. Güell, E. Matto, J. M. Lus, J. Poater, and M. Solà, J. Phys. Chem. A, 006, (submtted). E. Matto, M. Solà, P. Salvador, and M. Duran, Faraday Dscussons, 006, (accepted). J. Poater, X. Fradera, M. Duran, and M. Solà, Chem. Eur. J., 003, 9, 400. J. Poater, X. Fradera, M. Duran, and M. Solà, Chem. Eur. J., 003, 9, 3. E. Matto, M. Duran, and M. Solà, J. Chem. Phys., 005,, E. Matto, P. Salvador, M. Duran, and M. Solà, J. Phys. Chem. A, 006, 0, 508. J. Poater, M. Duran, M. Solà, and B. Slv, Chem. Rev., 005, 05, 39.