勢能積分的計算公式

*2

??

?由公式得到兩個java計算方法

VabA( double R ,int zn, double za,double zb, int n1,int L1 ,int m1 ,int n2 ,int L2 ,int m2 )

VabB( double R ,int zn, double za,double zb, int n1,int L1 ,int m1 ,int n2 ,int L2 ,int m2 )

zn是原子核的核電荷數，za，zb分別是ζ，ζ‘。公式里是假設兩個原子的核電荷數都是1。

驗算這兩個方法，參考了兩篇文章

1.Calculation of the One-Electron Two-Center Integrals Over Slater-Type Orbitals by Means of the Ellipsoidal Coordinates Method by SIDI MOHAMED MEKELLECHE, ABDELLATIF BABA-AHMED（1996）

2..Evaluation of Two-center One- and Two-electron Integrals over Slater Type Orbitals by YAKAR, Yusuf , ?ZMEN, Ayhan，ATAV, ülfet （2006）

得到表格

	n	l	n'	l'	λ	ζ	ζ'	R	YAKAR, Yusuf	計算值	計算值/參照值
zna=1	2	1	3	2	1	4.2	4.4	2.5	0.011498097921007800	0.011498097921203000	1.000000000016980000
	3	2	2	1	0	2.3	2.3	2.5	0.154172007331042000	0.154172007331022000	0.999999999999870000
	3	2	3	2	2	5.8	4.5	0.2	-1.471238017764440000	-1.471238018706040000	1.000000000640010000
	3	2	3	2	1	2	4.4	8.7	0.000000857400767270	0.000000857431091410	1.000035367522470000
	4	3	5	2	1	1.5	0.5	15	0.002572464174355450	0.002572464177822200	1.000000001347640000
	5	3	6	4	3	1.5	1	0.1	0.006949396708185720	-270.000000000000000000	?
	4	3	8	5	2	3.6	3.7	0.1	-0.007197120074513900		0.000000000000000000
	6	4	12	4	4	3.5	0.5	20	-0.000003811906504442		0.000000000000000000
	5	3	8	5	3	2.5	2	2.2	-0.182136029883179000		0.000000000000000000
	10	5	5	4	2	0.3	0.1	10	-0.006308043501843390		0.000000000000000000
	15	12	10	8	3	4	2.5	1.5	-0.034376370777494700		0.000000000000000000
	22	20	24	18	15	1.5	3.5	0.8	-0.001059509498481520		0.000000000000000000
											?
									SIDI MOHAMED MEKELLECHE	計算值	?
zna=35	3	2	3	2	0	6.5197	2.0387	3.75803	-0.486277949158820000	-0.486277948407194000	0.999999998454328000
	3	2	3	2	1	6.5197	2.0387	3.75803	0.265448562634640000	0.265448562222838000	0.999999998448656000
	3	2	3	2	2	6.5197	2.0387	3.75803	-0.041387530907970000	-0.041387530988898000	1.000000001955370000
											?
znb=17	3	2	3	2	0	6.5197	2.0387	3.75803	-0.086569711360165000	-0.086569711326935500	0.999999999616153000
	3	2	3	2	1	6.5197	2.0387	3.75803	0.039704646900045000	0.039704646872093500	0.999999999296014000
	3	2	3	2	2	6.5197	2.0387	3.75803	-0.005394666015387200	-0.005394666000166480	0.999999997178561000

結果表明用這個程序計算勢能積分，主量子數不能大于5.

代入比如第一組

double d2= VabA( 2.5 , 1 , 4.2, 4.4 , ? 2 , 1 , 1 , 3 , 2 , 1 ); // 0.011498097921203

最后一組

double d2= VabB( 3.75803 ?, 17 , 6.5197 , 2.0387 , ? 3 , 2 , 2 , 3 , 2 , 2 );// -0.00539466600016648

對應的java程序為

public static double VabA( double R ,int zn, double za,double zb, int n1,int L1 ,int m1 ,int n2 ,int L2 ,int m2 ) throws IOException, ParseException, InterruptedException {double f1=-zn* (2*za/ Math.pow(2*n1*(2*n1-1),0.5 ) )*HFoverlapA .Sab ( R, za, zb , n1-1, L1 , m1 ,n2 ,L2 ); return f1;}public static double VabB( double R ,int zn, double za,double zb, int n1,int L1 ,int m1 ,int n2 ,int L2 ,int m2 ) throws IOException, ParseException, InterruptedException {double f1=-zn* (2*zb/ Math.pow(2*n2*(2*n2-1),0.5 ) )*HFoverlapA .Sab ( R, za, zb , n1, L1 , m1 ,n2-1 ,L2 ); return f1;}

總結

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