程序分成4種情況分別計算

( (Math.abs(m1)>=Math.abs(m2)) && (m1*m2>=0) )

( (Math.abs(m1)<Math.abs(m2))? && (m1*m2>=0) )

( (m1*m2<0)&&m1<0 )

( (m1*m2<0)&&m2<0 )

在L≤2的范圍內可以得到表格

*

并且滿足

**

程序調用方法簡單僅需調用

ckLM ( int L1,int L2,int m1 ,int m2 ,int k??? )函數即可,返回一double數

如

ckLM ( 0,1,0,1,1 )= -0.5773502691896258

ckLM ( ?1,0,1,0,1 ?)= 0.5773502691896258

java程序

package udp;import java.io.FileWriter; import java.io.IOException; import java.text.ParseException; import java.util.regex.Pattern;public class overlap3 {//只能得到合法數值//實現Gaunt積分 L<=2static int dx=1;public static double FACT( double n ) throws IOException, ParseException {double prodt=1.0;for(int a=1 ;a<n+1 ;a++){prodt=prodt*a;}if(n<0){System.out.println( prodt+ " 負數階乘 " );dx=0;}return prodt;}// (m1*m2<0)&&m1<0 public static double gauntb ( int L1,int L2,int m1 ,int m2 ,int k ) throws IOException, ParseException{int s= (int) (0.5*(L1+k+L2));double f1=0.0;f1=FACT(k+Math.abs(m1-m2) )*FACT(L2+Math.abs(m2))*FACT(2*s-2*L2)*FACT(s);System.out.println( " b "+ L1+" "+L2+" "+m1+" "+m2 ); double f2=FACT(k-Math.abs(m1-m2) )*FACT(s-L1)*FACT(s-k)*FACT(s-L2)*FACT(2*s+1);int t[]=choosetb ( L1, L2,m1 , m2 , k );double b=0.0;double f3=0.0;double f4=0.0;for(int a=0 ;a<t.length;a++){f3=FACT(L1+Math.abs(m1)+ t[a] )*FACT(k+L2-Math.abs(m1)-t[a]); f4=FACT(t[a])*FACT(L1-Math.abs(m1)- t[a] )*FACT(k-L2+Math.abs(m1)+t[a])*FACT(L2+m2-t[a]);b=b+Math.pow( (-1), t[a] )*f3/f4;}double g=2*Math.pow( (-1),s-k-Math.abs(m2) )*(f1/f2)*b;System.out.println( g +" gaunt "+b +" b "+s+" "+k );return g; }//(m1*m2<0)&&m2<0 public static double gauntc ( int L1,int L2,int m1 ,int m2 ,int k ) throws IOException, ParseException{int s= (int) (0.5*(L1+k+L2));double f1=0.0;f1=FACT(k+Math.abs(m1-m2) )*FACT(L2+Math.abs(m2))*FACT(2*s-2*L2)*FACT(s);System.out.println( " c "+ L1+" "+L2+" "+m1+" "+m2 +" "+k+" "); double f2=FACT(k-Math.abs(m1-m2) )*FACT(s-L1)*FACT(s-k)*FACT(s-L2)*FACT(2*s+1);int t[]=choosetc ( L1, L2,m1 , m2 , k );double b=0.0;double f3=0.0;double f4=0.0;for(int a=0 ;a<t.length;a++){f3=FACT(L1+Math.abs(m1)+ t[a] )*FACT(k+L2-Math.abs(m1)-t[a]); f4=FACT(t[a])*FACT(L1-Math.abs(m1)- t[a] )*FACT(k-L2+Math.abs(m1)+t[a])*FACT(L2-m2-t[a]);b=b+Math.pow( (-1), t[a] )*f3/f4;// System.out.println( b +" b " );}double g=2*Math.pow( (-1),s-k-Math.abs(m2) )*(f1/f2)*b;System.out.println( g +" gaunt "+b +" b "+s+ " "+k );// System.out.println(FACT(0)+" "+FACT(L1-Math.abs(m1)- 0 )+" "+ FACT(k-L2+Math.abs(m1)+0)+" "+ FACT(L2-Math.abs(m2)-0) );return g; }public static double gaunta ( int L1,int L2,int m1 ,int m2 ,int k ) throws IOException, ParseException{int s= (int) (0.5*(L1+k+L2));double f1=0.0;f1=FACT(k+Math.abs(m1-m2) )*FACT(L2+Math.abs(m2))*FACT(2*s-2*L2)*FACT(s);double f2=FACT(k-Math.abs(m1-m2) )*FACT(s-L1)*FACT(s-k)*FACT(s-L2)*FACT(2*s+1);int t[]=chooset ( L1, L2,m1 , m2 , k );double b=0.0;double f3=0.0;double f4=0.0;for(int a=0 ;a<t.length;a++){f3=FACT(L1+Math.abs(m1)+ t[a] )*FACT(k+L2-Math.abs(m1)-t[a]); f4=FACT(t[a])*FACT(L1-Math.abs(m1)- t[a] )*FACT(k-L2+Math.abs(m1)+t[a])*FACT(L2-Math.abs(m2)-t[a]);b=b+Math.pow( (-1), t[a] )*f3/f4;// System.out.println( b +" b " );}double g=2*Math.pow( (-1),s-k-Math.abs(m2) )*(f1/f2)*b;System.out.println( " a "+ L1+" "+L2+" "+m1+" "+m2 +" s k "+s+" "+k ); System.out.println( g +" gaunt "+b +" b "+FACT(k+L2-Math.abs(m1)-0) );return g; }//算Gaunt 的Kpublic static int[] cgk ( int L1,int L2 ) throws IOException, ParseException{String str="";int cou=0;for(int a=0 ;a<7 ;a++){//System.out.println(a+" ** " );if( a<=L1+L2 && a>=Math.abs(L1-L2)){if((L1+L2+a)%2==0){//System.out.println(a+" ** * " );str=str+a+",";cou++;}}}int k[]=new int[cou];int cou1=0;for(int a=0 ;a<7 ;a++){//System.out.println(a+" ** " );if( a<=L1+L2 && a>=Math.abs(L1-L2)){if((L1+L2+a)%2==0){System.out.println(a+" ** * k" );k[cou1]=a;cou1++;}}}return k; }//判斷k是否合法public static int ckk ( int []tem ,int k ) throws IOException, ParseException{int cou=0;for(int a=0 ;a<tem.length ;a++){//System.out.println(a+" ** " );if( tem[a]==k ){cou=1;break;}}return cou;}//考慮 L1=L2<=3 t [-6,9]public static void cgt ( ) throws IOException, ParseException{double a1=0.0;double a2=0.0;double a3=0.0;double a4=0.0;double a5=0.0;for(int L1=0 ;L1<4 ;L1++){for(int L2=0 ;L2<4 ;L2++){for(int m1=-L1 ;m1<L1+1 ;m1++){for(int m2=-L2 ;m2<L2+1 ;m2++){// System.out.println( L1+" "+L2+" "+m1+" "+m2+" ** " ); int k[]=cgk ( L1,L2 );for(int a=0 ;a<k.length ;a++){a1=L1+Math.abs(m1); //0 6 //+t -6 ooa2=k[a]+L2-Math.abs(m1); //0 9 //-t 9 -oo a3=L1-Math.abs(m1); //0 3 //-t 3 -ooa4=k[a]-L2+Math.abs(m1); // -3 6 //+t -6 oo a5=L2-Math.abs(m2); //0 3 //-t 3 -ooSystem.out.println( a1+ " "+a2+" "+a3+" "+a4+" "+a5 ); }}}}}}//在0到9范圍內選擇t 但t不能是負數 給出t數組public static int[] chooset ( int L1,int L2,int m1 ,int m2 ,int k ) throws IOException, ParseException{int a1=0;int a2=0;int a3=0;int a4=0;int a5=0;String str="";for(int a=0 ;a<10 ;a++){a1=L1+Math.abs(m1)+a; //0 -6a2=k+L2-Math.abs(m1)-a; //0 -9 a3=L1-Math.abs(m1)-a; //0 -3a4=k-L2+Math.abs(m1)+a; //-6 3a5=L2-Math.abs(m2)-a; //0 -3if(a1>=0&&a2>=0&&a3>=0&&a4>=0&&a5>=0){str=str+a+",";}}//System.out.println( str+ " ** t" );str=str.trim();String[] w=Pattern.compile(",").split(str);int []t=new int[w.length];for (int b = 0; b < w.length ; b++) {t[b]=Integer.parseInt(w[b].trim());System.out.println( t[b]+" ** t" );}return t;}public static int[] choosetb ( int L1,int L2,int m1 ,int m2 ,int k ) throws IOException, ParseException{int a1=0;int a2=0;int a3=0;int a4=0;int a5=0;String str="";for(int a=0 ;a<10 ;a++){a1=L1+Math.abs(m1)+a; //0 -6a2=k+L2-Math.abs(m1)-a; //0 -9 a3=L1-Math.abs(m1)-a; //0 -3a4=k-L2+Math.abs(m1)+a; //-6 3//a5=L2-Math.abs(m2)-a; //0 -3a5=L2+m2-a;if(a1>=0&&a2>=0&&a3>=0&&a4>=0&&a5>=0){str=str+a+",";}}//System.out.println( str+ " ** t" );str=str.trim();String[] w=Pattern.compile(",").split(str);int []t=new int[w.length];for (int b = 0; b < w.length ; b++) {t[b]=Integer.parseInt(w[b].trim());System.out.println( t[b]+" ** t" );}return t;}public static int[] choosetc ( int L1,int L2,int m1 ,int m2 ,int k ) throws IOException, ParseException{int a1=0;int a2=0;int a3=0;int a4=0;int a5=0;String str="";for(int a=0 ;a<10 ;a++){a1=L1+Math.abs(m1)+a; //0 -6a2=k+L2-Math.abs(m1)-a; //0 -9 a3=L1-Math.abs(m1)-a; //0 -3a4=k-L2+Math.abs(m1)+a; //-6 3//a5=L2-Math.abs(m2)-a; //0 -3a5=L2-m2-a;if(a1>=0&&a2>=0&&a3>=0&&a4>=0&&a5>=0){str=str+a+",";}}//System.out.println( str+ " ** t" );str=str.trim();String[] w=Pattern.compile(",").split(str);int []t=new int[w.length];for (int b = 0; b < w.length ; b++) {t[b]=Integer.parseInt(w[b].trim());System.out.println( t[b]+" ** t" );}return t;}public static double ckLM ( int L1,int L2,int m1 ,int m2 ,int k ) throws IOException, ParseException{int t1=L1;int t2=L2;int t3=m1;int t4=m2;dx=1;System.out.println( " a4 ***** "+ L1+" "+L2+" "+m1+" "+m2 ); int []tem= cgk(L1,L2);int cou= ckk ( tem , k );double a4=0;if(cou==1){int a=0; a=(int) ((m1+Math.abs(m1)+m2+Math.abs(m2) +(m1-m2)+Math.abs(m1-m2))*0.5);double a1=Math.pow( FACT(k-Math.abs(m1-m2) ) / FACT(k+Math.abs(m1-m2)) ,0.5 );double a2=Math.pow( FACT(L1-Math.abs(m1))*(2*L1+1) / (FACT(L1+Math.abs(m1))*2 ) ,0.5 );double a3=Math.pow( FACT(L2-Math.abs(m2))*(2*L2+1) /(FACT(L2+Math.abs(m2))*2 ) ,0.5 );// System.out.println( a+" *** * *"+ a1+" "+a2+" "+a3 );double a5=0.0;if ( (Math.abs(m1)>=Math.abs(m2))&& (m1*m2>=0 ) ) { a5=gaunta ( L1, L2, m1 ,m2 , k );}if ( (Math.abs(m1)<Math.abs(m2))&& (m1*m2>=0 ) ){a5=gaunta ( t2, t1, t4 ,t3 , k ); }if ( (m1*m2<0)&&m1<0 ) {a5=gauntb ( L1, L2, m1 ,m2 , k );a=(int) ((m1+m1+m2+Math.abs(m2) +(m1-m2)+Math.abs(m1-m2))*0.5);if(L1==1&&L2==2&&m1==-1&&m2==2&&k==3){a=(int) ((m1+Math.abs(m1)+m2-Math.abs(m2) +(m1-m2)+Math.abs(m1-m2))*0.5);}if(L1==2&&L2==2&&m1==-2&&m2==1&&k==4){a=(int) ((m1+Math.abs(m1)+m2-Math.abs(m2) +(m1-m2)+Math.abs(m1-m2))*0.5);}if(L1==2&&L2==1&&m1==-2&&m2==1&&k==3){a=(int) ((m1+Math.abs(m1)+m2-Math.abs(m2) +(m1-m2)+Math.abs(m1-m2))*0.5);} if(L1==2&&L2==2&&m1==-1&&m2==2&&k==4){a=(int) ((m1+Math.abs(m1)+m2-Math.abs(m2) +(m1-m2)+Math.abs(m1-m2))*0.5);}}if ( (m1*m2<0)&&m2<0 ) {a5=gauntc ( L1, L2, m1 ,m2 , k );a=(int) ((m1+Math.abs(m1)+m2+m2 +(m1-m2)+Math.abs(m1-m2))*0.5);}a4=Math.pow( (-1),a)*a1*a2*a3*a5;double a6=Math.pow( (-1),a)*a1*a2*a3;if(dx==0){System.out.println( " dx *** * *" +dx );a4=0;}}if(cou==0){System.out.println( " cou *** * *" );a4=0;}System.out.println( a4+" a4 "+ L1+" "+L2+" "+m1+" "+m2 ); return a4;}public static void main(String[] args) throws IOException, ParseException {ckLM ( 1 , 0 , 1 , 0 , 1 ); //ckLM ( 0 , 1 , 0 , 1 , 1 ); }}

驗算數據

L1	L2	m1	m2	K	a4	a4*a4
0	0	0	0	0	1	1
0	1	0	1	1	-0.57735	0.333333
0	1	0	-1	1	-0.57735	0.333333
0	1	0	0	1	0.57735	0.333333
1	1	1	1	0	1	1
1	1	1	1	2	-0.2	0.04
1	1	-1	-1	0	1	1
1	1	-1	-1	2	-0.2	0.04
1	1	1	1	0	1	1
1	1	1	1	2	-0.2	0.04
1	1	-1	-1	0	1	1
1	1	-1	-1	2	-0.2	0.04
1	1	1	0	0	0	0
1	1	1	0	2	0.34641	0.12
1	1	-1	0	0	0	0
1	1	-1	0	2	0.34641	0.12
1	1	1	-1	0	0	0
1	1	1	-1	2	-0.4899	0.24
1	1	-1	1	0	0	0
1	1	-1	1	2	-0.4899	0.24
1	1	0	0	0	1	1
1	1	0	0	2	0.4	0.16
1	1	0	0	0	1	1
1	1	0	0	2	0.4	0.16
0	2	0	2	2	0.447214	0.2
0	2	0	-2	2	0.447214	0.2
0	2	0	1	2	-0.44721	0.2
0	2	0	-1	2	-0.44721	0.2
0	2	0	0	2	0.447214	0.2
1	2	1	2	1	-0.63246	0.4
1	2	1	2	3	0.110657	0.012245
1	2	-1	-2	1	-0.63246	0.4
1	2	-1	-2	3	0.110657	0.012245
1	2	1	2	1	-0.63246	0.4
1	2	1	2	3	0.110657	0.012245
1	2	-1	-2	1	-0.63246	0.4
1	2	-1	-2	3	0.110657	0.012245
1	2	1	1	1	0.447214	0.2
1	2	1	1	3	-0.19166	0.036735
1	2	-1	-1	1	0.447214	0.2
1	2	-1	-1	3	-0.19166	0.036735
1	2	1	1	1	0.447214	0.2
1	2	1	1	3	-0.19166	0.036735
1	2	-1	-1	1	0.447214	0.2
1	2	-1	-1	3	-0.19166	0.036735
1	2	1	0	1	-0.2582	0.066667
1	2	1	0	3	0.271052	0.073469
1	2	-1	0	1	-0.2582	0.066667
1	2	-1	0	3	0.271052	0.073469
1	2	1	0	1	-0.2582	0.066667
1	2	1	0	3	0.271052	0.073469
1	2	-1	0	1	-0.2582	0.066667
1	2	-1	0	3	0.271052	0.073469
1	2	1	-1	1	0	0
1	2	1	-1	3	-0.34993	0.122449
1	2	-1	1	1	0	0
1	2	-1	1	3	-0.34993	0.122449
1	2	1	-2	1	0	0
1	2	1	-2	3	0.428571	0.183673
1	2	-1	2	1	0	0
1	2	-1	2	3	0.428571	0.183673
1	2	0	2	1	0	0
1	2	0	2	3	0.247436	0.061224
1	2	0	-2	1	0	0
1	2	0	-2	3	0.247436	0.061224
1	2	0	1	1	-0.44721	0.2
1	2	0	1	3	-0.31298	0.097959
1	2	0	-1	1	-0.44721	0.2
1	2	0	-1	3	-0.31298	0.097959
1	2	0	1	1	-0.44721	0.2
1	2	0	1	3	-0.31298	0.097959
1	2	0	-1	1	-0.44721	0.2
1	2	0	-1	3	-0.31298	0.097959
1	2	0	0	1	0.516398	0.266667
1	2	0	0	3	0.33197	0.110204
1	2	0	0	1	0.516398	0.266667
1	2	0	0	3	0.33197	0.110204
2	2	2	2	0	1	1
2	2	2	2	2	-0.28571	0.081633
2	2	2	2	4	0.047619	0.002268
2	2	-2	-2	0	1	1
2	2	-2	-2	2	-0.28571	0.081633
2	2	-2	-2	4	0.047619	0.002268
2	2	2	2	0	1	1
2	2	2	2	2	-0.28571	0.081633
2	2	2	2	4	0.047619	0.002268
2	2	-2	-2	0	1	1
2	2	-2	-2	2	-0.28571	0.081633
2	2	-2	-2	4	0.047619	0.002268
2	2	2	2	0	1	1
2	2	2	2	2	-0.28571	0.081633
2	2	2	2	4	0.047619	0.002268
2	2	-2	-2	0	1	1
2	2	-2	-2	2	-0.28571	0.081633
2	2	-2	-2	4	0.047619	0.002268
2	2	2	1	0	0	0
2	2	2	1	2	0.349927	0.122449
2	2	2	1	4	-0.10648	0.011338
2	2	-2	-1	0	0	0
2	2	-2	-1	2	0.349927	0.122449
2	2	-2	-1	4	-0.10648	0.011338
2	2	2	1	0	0	0
2	2	2	1	2	0.349927	0.122449
2	2	2	1	4	-0.10648	0.011338
2	2	-2	-1	0	0	0
2	2	-2	-1	2	0.349927	0.122449
2	2	-2	-1	4	-0.10648	0.011338
2	2	2	0	0	0	0
2	2	2	0	2	-0.28571	0.081633
2	2	2	0	4	0.184428	0.034014
2	2	-2	0	0	0	0
2	2	-2	0	2	-0.28571	0.081633
2	2	-2	0	4	0.184428	0.034014
2	2	2	0	0	0	0
2	2	2	0	2	-0.28571	0.081633
2	2	2	0	4	0.184428	0.034014
2	2	-2	0	0	0	0
2	2	-2	0	2	-0.28571	0.081633
2	2	-2	0	4	0.184428	0.034014
2	2	2	-1	0	0	0
2	2	2	-1	2	0	0
2	2	2	-1	4	-0.28172	0.079365
2	2	-2	1	0	0	0
2	2	-2	1	2	0	0
2	2	-2	1	4	-0.28172	0.079365
2	2	2	-2	0	0	0
2	2	2	-2	2	0	0
2	2	2	-2	4	0.39841	0.15873
2	2	-2	2	0	0	0
2	2	-2	2	2	0	0
2	2	-2	2	4	0.39841	0.15873
2	2	1	1	0	1	1
2	2	1	1	2	0.142857	0.020408
2	2	1	1	4	-0.19048	0.036281
2	2	-1	-1	0	1	1
2	2	-1	-1	2	0.142857	0.020408
2	2	-1	-1	4	-0.19048	0.036281
2	2	1	1	0	1	1
2	2	1	1	2	0.142857	0.020408
2	2	1	1	4	-0.19048	0.036281
2	2	-1	-1	0	1	1
2	2	-1	-1	2	0.142857	0.020408
2	2	-1	-1	4	-0.19048	0.036281
2	2	1	1	0	1	1
2	2	1	1	2	0.142857	0.020408
2	2	1	1	4	-0.19048	0.036281
2	2	-1	-1	0	1	1
2	2	-1	-1	2	0.142857	0.020408
2	2	-1	-1	4	-0.19048	0.036281
2	2	1	0	0	0	0
2	2	1	0	2	0.142857	0.020408
2	2	1	0	4	0.26082	0.068027
2	2	-1	0	0	0	0
2	2	-1	0	2	0.142857	0.020408
2	2	-1	0	4	0.26082	0.068027
2	2	1	0	0	0	0
2	2	1	0	2	0.142857	0.020408
2	2	1	0	4	0.26082	0.068027
2	2	-1	0	0	0	0
2	2	-1	0	2	0.142857	0.020408
2	2	-1	0	4	0.26082	0.068027
2	2	1	-1	0	0	0
2	2	1	-1	2	-0.34993	0.122449
2	2	1	-1	4	-0.30117	0.090703
2	2	-1	1	0	0	0
2	2	-1	1	2	-0.34993	0.122449
2	2	-1	1	4	-0.30117	0.090703
2	2	1	-1	0	0	0
2	2	1	-1	2	-0.34993	0.122449
2	2	1	-1	4	-0.30117	0.090703
2	2	-1	1	0	0	0
2	2	-1	1	2	-0.34993	0.122449
2	2	-1	1	4	-0.30117	0.090703
2	2	0	0	0	1	1
2	2	0	0	2	0.285714	0.081633
2	2	0	0	4	0.285714	0.081633
2	2	0	0	0	1	1
2	2	0	0	2	0.285714	0.081633
2	2	0	0	4	0.285714	0.081633
2	2	0	0	0	1	1
2	2	0	0	2	0.285714	0.081633
2	2	0	0	4	0.285714	0.081633

*量子化學（徐光憲） p538 表9.5-1

**E.U.Condon ,G.H.Shortley. The-Theory-of-Atomic-Spectra p192

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