42:. When computers were first used to calculate bonding in molecules, it was only possible to calculate diatomic molecules. As computers advanced, it became possible to study larger molecules, but the use of this approximation has always allowed the study of even larger molecules. Currently semi-empirical methods can be applied to molecules as large as whole proteins. The approximation involves ignoring certain integrals, usually two-electron repulsion integrals. If the number of orbitals used in the calculation is N, the number of two-electron repulsion integrals scales as N. After the approximation is applied the number of such integrals scales as N, a much smaller number, simplifying the calculation.
492:
266:
667:
487:{\displaystyle \langle \mu \nu |\lambda \sigma \rangle =\iint \left(\mathbf {\chi } _{\mu }^{A}(1)\right)^{*}\left(\mathbf {\chi } _{\nu }^{C}(2)\right)^{*}{\frac {1}{r_{12}}}\mathbf {\chi } _{\lambda }^{B}(1)\mathbf {\chi } _{\sigma }^{D}(2)d\tau _{1}\,d\tau _{2}\ }
216:
753:
564:
580:
901:
121:
258:
80:
129:
863:
It is possible to partly justify this approximation, but generally it is used because it works reasonably well when the integrals that remain –
675:
35:
500:
860:, i.e. when the basis functions for the first electron are on the same atom and the basis functions for the second electron are the same atom.
938:
774:
800:
use the zero differential overlap approximation completely. Methods based on the intermediate neglect of differential overlap, such as
793:
662:{\displaystyle \langle \mu \nu |\lambda \sigma \rangle =\delta _{\mu \lambda }\delta _{\nu \sigma }\langle \mu \nu |\mu \nu \rangle }
832:, i.e. when all four basis functions are on the same atom. Methods that use the neglect of diatomic differential overlap, such as
965:
866:
89:
228:
211:{\displaystyle \mathbf {\Phi } _{i}\ =\sum _{\mu =1}^{N}\mathbf {C} _{i\mu }\ \mathbf {\chi } _{\mu }^{A}\,}
53:
781:
700:
922:
944:
934:
39:
31:
777:
926:
914:
837:
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17:
841:
497:
The zero differential overlap approximation ignores integrals that contain the product
959:
930:
748:{\displaystyle \delta _{ij}={\begin{cases}0&i\neq j\\1&i=j\ \end{cases}}}
948:
559:{\displaystyle \mathbf {\chi } _{\mu }^{A}(1)\mathbf {\chi } _{\nu }^{B}(1)}
260:
are coefficients, the two-electron repulsion integrals are then defined as:
797:
813:
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833:
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741:
896:{\displaystyle \langle \mu \mu |\lambda \lambda \rangle }
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92:
56:
769: / 2) from / 2 (approximately
895:
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661:
558:
486:
252:
225:is the atom the basis function is centred on, and
210:
115:
74:
757:The total number of such integrals is reduced to
788:Scope of approximation in semi-empirical methods
773: / 8), all of which are included in
765: + 1) / 2 (approximately
116:{\displaystyle \mathbf {\chi } _{\mu }^{A}\ }
8:
921:. Chichester: John Wiley and Sons. pp.
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34:theory that is the central technique of
917:Introduction to Computational Chemistry
253:{\displaystyle \mathbf {C} _{i\mu }\ }
75:{\displaystyle \mathbf {\Phi } _{i}\ }
794:Pariser–Parr–Pople method
30:is an approximation in computational
7:
25:
903: – are parameterised.
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1:
844:, also do not apply it when
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50:If the molecular orbitals
82:are expanded in terms of
28:Zero differential overlap
18:Zero-differential overlap
46:Details of approximation
966:Computational chemistry
931:2027/uc1.31822026137414
782:post-Hartree–Fock
913:Jensen, Frank (1999).
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663:
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36:semi-empirical methods
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816:do not apply it when
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792:Methods such as the
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940:978-0-471-98085-8
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574:. This leads to:
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112:
86:basis functions,
71:
40:quantum chemistry
32:molecular orbital
16:(Redirected from
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570:is not equal to
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784:calculations.
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97:
66:
61:
47:
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24:
14:
13:
10:
9:
6:
4:
3:
2:
978:
967:
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946:
942:
936:
932:
928:
924:
919:
918:
911:
910:
906:
904:
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876:
873:
861:
859:
856: =
855:
851:
848: =
847:
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839:
835:
831:
828: =
827:
824: =
823:
820: =
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764:
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167:
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126:
125:
124:
105:
100:
95:
85:
64:
45:
43:
41:
37:
33:
29:
19:
916:
862:
857:
853:
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845:
829:
825:
821:
817:
791:
770:
766:
762:
758:
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671:
571:
567:
496:
222:
220:
83:
49:
27:
26:
907:References
796:(PPP) and
949:466189317
891:⟩
888:λ
885:λ
877:μ
874:μ
871:⟨
775:ab initio
712:≠
681:δ
657:⟩
654:ν
651:μ
643:ν
640:μ
637:⟨
632:σ
629:ν
625:δ
619:λ
616:μ
612:δ
605:⟩
602:σ
599:λ
591:ν
588:μ
585:⟨
538:ν
533:χ
512:μ
507:χ
473:τ
459:τ
436:σ
431:χ
410:λ
405:χ
381:∗
356:ν
351:χ
338:∗
313:μ
308:χ
297:∬
291:⟩
288:σ
285:λ
277:ν
274:μ
271:⟨
243:μ
198:μ
193:χ
183:μ
157:μ
153:∑
136:Φ
101:μ
96:χ
60:Φ
960:Category
947:
937:
798:CNDO/2
736:
672:where
572:ν
568:μ
566:where
482:
248:
221:where
188:
146:
111:
70:
925:–82.
814:SINDO
810:ZINDO
806:MINDO
945:OCLC
935:ISBN
852:and
840:and
834:MNDO
812:and
802:INDO
780:and
123:as:
927:hdl
842:AM1
838:PM3
38:in
962::
943:.
933:.
923:81
836:,
808:,
804:,
396:12
951:.
929::
881:|
858:D
854:C
850:B
846:A
830:D
826:C
822:B
818:A
771:N
767:N
763:N
761:(
759:N
733:j
730:=
727:i
722:1
715:j
709:i
704:0
698:{
693:=
688:j
685:i
647:|
608:=
595:|
554:)
551:1
548:(
543:B
528:)
525:1
522:(
517:A
477:2
469:d
463:1
455:d
452:)
449:2
446:(
441:D
426:)
423:1
420:(
415:B
392:r
388:1
376:)
372:)
369:2
366:(
361:C
345:(
333:)
329:)
326:1
323:(
318:A
302:(
294:=
281:|
240:i
235:C
223:A
203:A
180:i
175:C
168:N
163:1
160:=
149:=
141:i
106:A
84:N
65:i
20:)
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