Indi an Journal of Pure & Applied Phys ics Vol. 39, May 200 1, pp. 275-28 1

Generalised effective core potential for atomic Pd and A~ _ - -- J J

R PjSemw~1 & A Sharma

/ SGR-; ('PG) College, Dehradun 248 000 Received 8 November 2000; accepted 8 January 2001

G.ransition energies have bee~ calcu lated for ;d ~ndlC~g atoms with effecti ve potentials. in the framework ~f Hartrec­Fock approx imation over the conflgurallon [--]4.1' 41' 4d . The resu lt gives Improvemen t ot the reproductIOn ot translllOIl energies from all electron calcul ati ons compared with standard e ffect ive core potential method. \ '" l -( I

) 1 Introduction

Effective core potential (ECP) approach for shape consistent pseudoorbitals is most acti ve ly used in calculations of e lectronic structure of molecules with heavy atoms l.2.3 But accuracy of the calculations with standard Kohn-type ECP is essentially limited with the demand that onl y node less pseudoorbitals must be used in generati on of ECP components. Moreover, for system contallllllg tranSttlOn meta ls, the presence of outermost node in the 45 orbitals may play an important ro le in c he mical bond. In the cases of a deep reorgani zati on of electronic confi gurati on structure of a va lence region, when a change of energy is comparable with the energy of the va lence electrons of considered atomic fragment, we wi II take into account explic itly (but not by means of the ECP approach) interacti ons with outer most core electrons. It is important in calculati ons of molecules when studying the internuc lear di stances smaller than equilibrium ones and the transitions to high exc ited states . It was estimated in many test calcul ations that the accuracy of such generali zed ECP for transition ene rgies is in linear dependence on the volume of the smoothed region. But high accuracy and rather bad behaviour of shape of the genera li zed ECP put seri ous obs tac les for approximat ion of numerical potential s with Gaussian functions.

In thi s work, the authors represent the ir first results of Gaussian ex pansion of generali zed ECPs for Pb and Ag atoms in the framework of non­relativi stic approach in order to examine the abilities in compari son with a ll e lectron and standard ECP accuracy.

2 Scheme of Generation

The method of the ECP generati on which the authors used here was bas icall y suggested in works~ '

7 and was then developed for the case of pseudo­orbitals with nodes. Thi s method a ll ows to include both va lence and outermost core e lectrons in the ECP generation scheme. The authors have a lso made use of workx .~ . The ma in steps of thi s procedure are :

I) Node less pseudoorbitals and pseudo-orbitals with one node tji" /( r ) are deri ved from the numeri cal

Hartree-Fock (HF) outermost core and va lence orbitals 'V"tCr) respectively. These pseudo-orbitals sati sfy the following conditions:

k

fer) = r Y L ;=0

; air,

l = 0 , I , .. .... , L,

r :2: Rc ( I )

r< Rc

n = v, oc

where L is the one more than the highest moment of the core e lectrons, U and oc denote va lence and outer core electrons respect ive ly . The coeffic ients {/ j

are determined by the foll owing requirements:

• Iii", is normali zed

• f and its fi rst 3 (or 4) deri vat ives match \V!!l

and its derivatives

• f is node less

• liivl and V//lc.I are orthogonal ized

• 'V"I is a smooth pseudo.orbital

276 INDIAN J PURE & APPL PHYS , VOL 39, MAY 2001

\jI1l1 ensure the enough smooth shape of the potentia l.

Rc is chosen near such extremum of the orbita l that the corresponding pseudoorbita l has the above defined number of nodes. It should be emphasized that radii Rc for the different orbita ls mu st be chosen closely to each othe r. Here, y is chosen near L+ I to ensure enough repul sion of the pseudoorbita ls from a core region .

2) Numerical potentia ls UIlI are deri ved for eac h 1=0, ..... .. , L for va lence and outer core pseudo-

orbitals so that the \jill I are soluti ons of the HF equati on for a pseudo-ato m with removed inner core orbita ls,

_ [I r 2 Z /(1- 1) ] _ U"I(r) =V/nrl ( r ) --) +--:-- - )-- J(r)+K(r)+E ,,1 V/"I (r) . .. (2)

2 dr- , 2r-

where, Z is the atomic charge, ; and K are the Coul omb and exchange operators calcul ated with

the pseudo-orbita ls, and till is the orbita l energy of

corresponding orbita l \jI1l1' In the case of the pseudo­orbita l w ith a node the potenti a l is s ingul ar because the nodes of denominator and numerator do not co incide. However, in practice these nodes are c lose to each other and the most appropri ate soluti on of thi s probl em is an interpolation of the potential in vic inity of the pseudo-orbital node. Error in an orbi ta l energy reproducti on due to thi s in terpo lation is small enough (because the pseudo-functi on is small in vic inity of node) and does not exceed the e rrors of the ECP method connected with the orbita l smoothing and the approx imati on of in teraction with inner core e lectrons. T hen ECP is written as a Hormiti an operator in the fo rm:

L· I I

U =(}VL(r)+ L,.U"/(r)-UvL (r)L,. III11 ><lIIII+

1=0 /II - I

L

+ L,.{ [UOC.l( r )+U,·/(r)]POc,/ + POCA UOc. /(r)-Uvl(r)]- i10 C. ,·.J'oC.1 }

1=0

L

POC.I = L,.I OC,I.II, >< OC.I.IIII

11I~-1

... (3)

. . . (4)

.. . (5)

where Ilm) (1m I is the projector on the sphe rica l

harmoni c Y I/II and I DC, I , m) (oc,l ,m I is the radi a l

projec tor onto the oute r core pseudoorbita l I / r \j/ ,'1'

Y I/II'

3) The numeri cal potentials a re then fined ana lytically by Gauss ian funct ions (fo ll ow ing work' in the fo rm :

[ Nfl ] :2 I L VI'L( r ) - ,-. - r. = ,

{ Uvl(r)-V vdr) } r 2

{ Uoc.! (r ) - U,'/(r)}/,

n. = v

1= 0, . . . . ... , L-I , I/, = V

I= O, .... . .. , L ,I1 = oe

... (6)

where Ne is the number o f core e lec trons. T he Gauss ian fun ctions have the form:

... (7)

where Il = 0, I or 2. The numerica l pseudoorbita ls are fitted to Gauss ian orbi ta ls of the fo rm:

. . . (8)

where Ni is the standard normali sati on bctor. T hL:

linear (C;) and non-linear (ai) paramete rs are optimi sed uSll1 g Marqua rdt least-squ ares procedure 111, which was modi fied in this work. It should be emphas ized that the approx imati on accuracy of potentia ls and pseudo- rbital s by Gauss ian functions must be rather high because the accuracy of genera li zed ECP method is la rger than standa rd ECP method.

2.1 Analytical approximation of potentials and pseudoorbitals

The analyt ical fits to the numerica l potenti a ls fo r the atoms Pd and Ag are g iven in Tables 1-4 w he re

the parameters Ili, a i and C, are defined in Eg. (7) . In these tables the outer-core potenti a ls (e.g. 3S-4S potenti a l) refer to (UOC,I- U rI) ,) and the va lence potentia ls (e .g. 4S -4f potenti a l or 4 f potenti a l) refe r to ((U,r U"L) r2 for I = 0 , . ... . . , L-I or C(U,L-Na/r)/ for l = L as in Eg. (6).

Gauss ian bas is sets to describe the oLlte r core and va lence pseudo-orbita ls of the ato ms Pd and Ag a re compiled in Tables 1-4 as derived from the leas t square fits to the pseudo-orbita ls. T he parameters C, and ai in these tab les are defined in Eg. (8) where

non-linear parameters ai for the oute r-core and va lence pseudo-orb ita ls are be li eved to be the same.

SEMW AL & SHARMA : EFFECTIVE CORE POTENTIAL FOR Pd & Ag 277

Table I - Effective core potenti al fo r atom Pd

I1 j U j Cj Ili Uj Ci

4s-4d potential 3s-4s potential

0 82.5400 149 3.0000000 I 1540.0329682 34. 11 4979 1

64 1.8996497 - 55.70 19 112 2 1805.8905368 - 4886.9908702

397.3 1680 19 110.0 177843 2 1058.82 16935 3536.9867593

205.7362939 - 1709.5 186795 2 620.8036 178 -3 117.29469]2

141. 1296626 6 195.4939745 2 363.9868 112 2036.60 18882

114.5738096 -45 10.4289 105 2 2 13.4 111254 - 1684.4 180936

2 134.267 15 19 - 17553 .960 11 29 2 125. 1262602 1223 .7 10438 1

2 82.7J69 120 63 11t .485 176 1 2 73.36347 14 - 1203.3763545

2 67.0583256 -8 1029. 1865637 2 43.0 14 1437 502.4932044

2 53.8267896 45559.8748965 2 25.2 1986 11 - 111.66369 16

2 42.0934844 - I 03 1 I .3 2 1 5407 2 14.7867966 20.30 1998R

2 24.6774938 486.792769] 2 8.6697287 - 1.964971 7

2 13.7485587 -49.809605R

2 8. 1049 13 1 8.2045293

2 4.9844569 - 0.9992 132

0 92.0255125 2.0000000 I 3300.8808904 -85.76082 14

509.4060 1 00 - 16 1.1 690252 2 3738.3865080 9733.6755363

355. 1506596 426.996 1604 2 2 11 6.9400 150 - 5 187.2938533

256.40484 14 -236. 1395493 2 11 98.76 18234 3063.805 1075

2 3 15.2597474 -3095 .9398202 2 678.824 1042 - 1987.68 1 ]902

2 187.6 105386 8022.7284720 2 ]84.]984313 1066.52401 07

2 129.2847903 - 14528.4896395 2 2 17.6736994 -820.0427432

2 93.7728239 20683.2422059 2 123.2623 12 1 524.253] 73 1

2 69.9217566 - 19457.2047514 2 69.7998776 - 492.304077.1

2 53.0353624 1077 1.9360434 2 39.5256493 203 .5 1426 1()

2 39. I 25425R -2593. 5795334 2 22 .3822305 -4 1.7028432

2 26.0064962 274.7930099 2 12.6744089 5.6594800

2 16.64 11 660 - 17.8727919

R4468.5495824 -2.6903492 97.R436475 0.084847R

33649.4225501 - 0.84]7707 2 127.6946749 - 11.09 160%

I 13468.35273 10 -5.37 16772 2 84.2 147273 30007723 1

2 10770.5402638 -229.5680 156 2 55 .5927350 -5 1.4950644

2 4423.5770745 - 380.6X34730 2 36.7010705 6 1.025457()

2 1738.8(>92353 - 196.12:19799 2 24. 1897641 - 54.28850%

2 702.5325786 - 12:1.9859928 2 15.9226577 4 1.2657704

2 272.9951770 - 67.4307387 2 10.47685 1 1 -29.340292X

2 122.5336792 -34.2749285 2 6 .8t)3282 5 18.3 815001

2 45 .0920142 -75607% 7 2 4.5356586 - t). 054376')

2 18.4347650 0.3320059 2 2.9846498 3. 1844966

2 7.4257626 -0.0570035 2 1.964 1427 - 0.5763411

278 INDlAN J PURE & APPL PHYS , VOL 39, MAY 2001

Table 2 : Effective core potential for ion Ag+

II i Uj Ci Il j Uj c-I

4s-4d potenti al 3s-4s potenti al

0 99.345 153 1 3.0000000 2426.5465705 -79 .0672663

1304.1656961 - 143.577537 1 2888.8460061 8467 .3022049

881.6277659 387.7794870 2 1719.6107728 -6091.6766 130

6 19.5521600 -2 11.0260462 2 1023.6133057 4456.0 I 00806

2 821.981 1473 -3949.995573 1 2 609.3 14745 1 -3442.0205359

2 477.7410682 6543 .950 1389 2 362.6999 146 2 162.945651 I

2 295. 1183428 - 7496. 1322003 2 215.9002865 - 1796. 259 1502

2 176.0129475 17520.2040771 2 128.5 165280 1304.85592 15

2 117.8472505 -59805 .3792543 2 76.5005838 - 1275.8389547

2 92.8749717 108022.7396597 2 45.5376395 52307 143 11

2 73.4617503 - 103538.3849660 2 27. 1066770 - 10 1 .488 I 5 1 8

2 58.5726404 55814.9442695 16. 1354858 12.362758 1

2 46.7500156 -134 10.52 10449

2 29.9 176170 593.56742 11

2 17.7175814 -2 1.7746354

0 92.6334067 2.0000000 3377.4090464 -87.0952000

490.0404565 - 167.1354325 2 3832.540 1908 10001 .0875992

350.6989792 435.3579598 2 2 174.50 183 15 - 53(,9 02 14862

I 264.5 180244 -236.8734988 2 1233 .766 11 32 3 179.888344 1

2 3 12.57 17964 - 2509.3 188335 2 700.0 126651 -2076.7426505

2 165.8684138 11 375 .3754994 2 397.1723052 III :U784755

2 11 8.768 1842 -37089.8042997 2 225.347 1228 -860.2054289

2 93.7288642 61626.5676872 2 127.857 1670 547.0584754

2 73 .3288006 -59 128.5300598 2 72.5434386 -5 11.7754 136

2 58.3648238 34705.732 1878 2 41 . 1596049 2 13.2445260

2 46.7964869 -9570. 177 1360 2 23.3530248 - 44.2340338

2 32.0836414 647.629473 1 2 13.2500439 6.0425liGlJ

2 19.6319787 -22.5340092

87043.5835885 - 1.437892 1 101.868 1191 0.08840:-19

384 1 1.0577262 - 0.0981609 2 133.9373806 - 11.4876987

16990.534403 1 - 1.5664554 2 88.05 12082 30.69 17 194

7504.0486842 -2.9880409 2 57.8853732 -52.5815904

33 I 5. 14085 I 5 -3.3 125476 2 38.054 1786 62.7114:-1306

2 2926.0623303 - 161 .83608 10 2 25.0 170372 -56. '203974

2 1299.68207 17 - 154.8794549 2 16.4463448 44.267667 1

2 579.23974 19 -96.91 8750 2 10.8 11 8222 - 32.294 1677

2 240.8863352 -60.3042800 2 7. 107820 I 20.6%48:14

2 11 9.040644 1 - 25 .8602540 2 4.6727220 - IO.39<)j550

2 50.4739608 - 7.8334570 2 3.0718745 3.7045li37

2 2 1.9599573 0.0263594 2 2.0 194680 -0.0759233

SEMW AL & SHARMA: EFFECTIVE CORE POTENTIAL FOR Pd & Ag 279

Table 3 - Pseudo-orbitals for atom Pd

c' · I Cj c · I Cj

U; 3s orbital 4s potential Ui 3p orbitals 4p orbital s 95.935 1977 -0. 11 42295 - 0.0 1907 14 179. 1429848 0.0 127320 0.0029 154 51.5538653 0.45338 12 0.1217274 71.76 15260 -0.04 14057 0.0000505 30.1 704696 -0.829 1909 -0.3702979 43.8040272 0.0936951 0.0323368 15.1193380 0.1382524 0.06 11 538 12000502 1 -0.5422740 - 0.3 188 153 7.2473300 0.7108 16 1 0.582 1004 5.3754311 -0.4988799 - 0.2574267 4.1055 130 0.4351855 0.5283382 204 11 241 -0.0635899 0.4839675 2.0744870 0.0708629 -0.300 1005 0.8710044 0.0053745 0.579433 1 1.1104 164 -0.0028 185 -0.70575 19 0.3362 166 -0.0017245 0. 1464544 0.5863618 0.0025764 -0.37 11 359 0.28523 16 -0.000 1559 - 0.0585907

3s orbital 4d potentia l 123.4745163 0.0647837 0.020 1626 37 .1 99426 1 0.2804705 0.0844923 37.847375 1 0.4927R84 0.153 1503 5.5947 175 0.34<)2355 0.0225006 2.0730 175 0.05595 18 -0.3495030 0.7909836 0.0031689 - 0.4708596 0.2838 116 0.00 14770 - 0.3256661 0.0949582 -0.0004292 - 0.1121115

Tab le 4 - Pseudo-orbital s for ion Ag+

Cj c· I Cj c · I

Ui 3s orbital 4s potential Ui 3p orbita ls 4p orbita ls 95.23 10039 -. 13 15962 - 0.0224591 180.5374 11 I 0.0134331 0.0032205 52.6070985 .5640299 0.1570792 7 1.9182024 -0.05 15843 - 0.0014356 33. 177073 1 -.9037203 -0.3934868 47. 1043163 0. 1054271 0.0349161 16.4037387 .09320 13 0.030290 1 12.7685536 - 0.5346597 - 0.3208867 7.8897439 .6928975 0.5754429 5.7882 184 - 0.5018370 - 0.269903 1 4.5419630 .4595247 0.546171 3 2.26494 10 - 0.0685822 0.46033 13 2.3577290 .0890031 -0. 1901224 .9893256 0.0050997 0.5899115 1.31630 12 - .0029540 -0.7042504 .4014888 -0.0017456 0. 16 14657 .689 1170 .0030889 -0.4539633 .336985 1 - .000186 1 -0070 1632

3d orbital 4c1 potential 150.0552815 .0504200 .0 166352 45.5809448 .2350569 .0774282 17.2 187777 .4640255 . 1539498 7.050 1092 .3988500 .0763 170 2.7900965 .09 17566 - .2744357 1.1415890 -.0020601 - .4756507 .4445921 .0019442 -.3662646 .1663763 -.0005652 -. !202807

280 INDIAN J PURE & APPL PHYS, VOL 39, MAY 2001

Table 5 - States averaged over configuration for Pd

Configuration Extract HF Standard ECP Our ECP 4s4p4d 3s3p3d4s4p4d

All Transition energy Absolute error Relative error Absolute error Relative error (in au) (in %) (in %)

4S24p~4d IU 0.0000 0.0000 0.00 0.0000 0.00 4S24p~4ds 0.9443 0.0048 0.5 1 0.0007 0.08 4s24p(,4d9 0.245 1 0.0022 0.90 0.0002 0. 10 4s24p54d9 2.8889 0.0149 0.52 0.00 13 0.05 4s24p54d lll 2.1723 0.0 129 0.60 0.00 10 0.04 4s14p(,4d9 4.1336 0.0 183 0.44 0.00 18 0.04 4s14p(,4d IU 3.4035 0.016 1 0.47 0.00 14 0.04 4s24p44d lO 4.8889 0.0274 0.56 0.0025 0.05 4s14p54d10 6.066 1 0.0275 0.45 0.0024 0.04

Table 6 - States averaged over configurat ion for Ag

Configuration Extract HF Standard ECP Our ECP 4s4p4d 3s3p3d4s4p4d

All Transition energy Absolute error Relative error Absolute error Relative error (in au) (in %) (i n %)

4s24p(,4d lO 0.0000 0.000 0.00 0.0000 0.00 4s24p(,4d95s I 0.2130 0.0049 2.28 0.0007 0.3 1 4s24p(,4d95p I 0.3760 0.0053 1.41 0 .0006 0 .1 7 4s24p(,4d94fl 0.6208 0.0051 0.82 0.0005 0.09 4s24p54d 105s I 2.2988 0.0 142 0.62 0.0011 0.05 4s24p54d 1114fl 2.711 5 0.0 147 0.54 0.00 10 0.03 4s24p(,4d lOS I -0.2 17 1 0.000 1 0.07 0.0001 0.05 4S24p~4d 105p I -0.1183 0.000 1 0.12 0.000 1 0.04 4s24p(,4d 1U5d l -0.0546 0.0000 0.00 0.0000 0.00

Table 7 - Terms of configuration 4s24p(,4dx fo r atom Pd

Terms Extract HF Standard ECP Our ECP 4s4p4d 3s3p3d4s4p4d

All Transition energy Abso lute error Relative error Absolute error Relative error (i n au) (i n %) (in %)

Average 0.0000 0.00000 0.00 0.0000 0.00 IS 0.17645 0.00666 3.78 0.00012 0.Q7 3p 0.02387 0.00 11 9 4 .98 0.00002 0.Q7 ID 0.0 1506 0.00046 3.02 0.00001 0.07

Table 8 - Terms of configuration 4s24p(,4d95pl for ion Ag+

Terms Extract HF Standard ECP OurECP 4s4p4d 3s3p3d4s4p4d

All Transition energy Absolute error Relative error Absolu te error Relative error (i n au) (in %) (in %)

Average 0.0000 0.000 0.00 0.00000 0.00 IS 0.01 370 0.00011 0.8 1 0.00007 0.53 3p -0.0 1719 -0.0003 1 1.80 0.00000 0.01 ID 0.01129 0.000 17 1.46 -0.00002 0. 14 3D 0.01 129 0.000 17 1.46 -0.00002 0. 14

SEMWAL & SHARMA: EFFECTIVE CORE POTENTIAL FOR Pd & Ag 281

3 Results and Conclusion

Both the all-electrons HF calculations and the HF calculations with ECP were performed using the atomic numerical non-re lativi stic programme, which was modified for calculations with ECP. In this programme, the authors used ECP-components that were expanded upon Gau ssian function s. The calculations were performed using both their scheme of the ECP generation and the standard scheme, where in the generation of ECP as well as in the calculations only valence electrons were included.

The transition energies calculated with e ffective potentials in the framework of HF approximation for wave functions provide a sensitive test on the ir applicability to molecular calculations. In Tables 5-8 the results of these calculations for atoms Pd and Ag are represented . These Tables 5-8 contain the numerical all-e lectron HF tran sition energies and the absolute and relative en'ors of their reproduction in calculations with ECP. Both standard and ECP were derived from the numerical HF wave function s for the states which have been averaged over the configurations 4S24p64dlO for Pd and Ag orbitals involved in SCF iterations are al so given in these Tables 5-8 . The choice or orbital s was based on the analysis of mean radii for each orbital to avoid large overlapping of removed and SCF orbitals and on the corresponding orbital energy analysis . These data show the significant improvement of the reproduction of the transition energies from all­electron calculations 111 the ECP method as

compared with the standard meth od.

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