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Names: »mchfex«
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C ------------------------------------------------------------------
C 2-1 E X A M P L E 1
C ------------------------------------------------------------------
C
C THE FIRST EXAMPLE ILLUSTRATES HOW DATA CAN BE SET UP FOR AN
C ISOELECTRONIC SEQUENCE. NOTE THAT FOR EACH ATOM OR ION, THE
C SLATER INTEGRALS AND RADIAL FUNCTIONS ARE PRINTED. THE LATTER
C ALSO ARE OUTPUT, A FUNCTION AT A TIME, FOR FUTURE INPUT.
C
5 5 0 0
6 6 7 0
BE 1S 4. 180 2 2 1
1S2/2S2
1S 1 0 0.0 1 0 0 2
2S 2 0 3.0 1 0 0 2
T 12 4
1 B+ 5.0
T 12 4
1 C+2 6.0
T 12 4
1 O+4 8.0
T 12 4
1 NA+7 11.0
T 12 4
1 P+11 15.0
T 12 4
1 CA+16 20.0
T 12 4
1 MN+21 25.0
T 12 4
1 ZN+26 30.0
T 12 4
*
C
C THOUGH RESULTS ARE MEANINGFUL PHYSICALLY ONLY FOR INTEGRAL
C VALUES OF Z, A USEFUL FEATURE WHEN DOING CALCULATIONS FOR
C NEGATIVE IONS WHERE INITIAL ESTIMATES ARE NOT WELL KNOWN, IS
C THE ABILITY TO TURN OFF THE NUCLEAR CHARGE IN FRACTIONAL STEPS,
C THEREBY APPROACHING THE RELATIVELY UNSTABLE STATE WITH GOOD
C INITIAL ESTIMATES.
C
C ------------------------------------------------------------------
C 2-2 E X A M P L E 2
C ------------------------------------------------------------------
C
C THE SECOND EXAMPLE ILLUSTRATES A FIXED CORE CALCULATION. IN THE
C FIRST CASE, HARTREE-FOCK CALCULATIONS ARE PERFORMED FOR THE NA+
C CORE. THEN, WITH THESE FUNCTIONS FIXED AND STILL IN MEMORY, A
C FIXED CORE CALCULATION IS PERFORMED FIRST FOR 3S AND THEN FOR 3P.
C IN THIS CASE NO CALCULATIONS ARE SUMMARIZED NOR ARE ANY RESULTS
C PUNCHED. NOTE ALSO THE ZERO OCCUPATION NUMBER FOR 3S IN THE
C CALCULATION FOR 3P. THIS IS ALLOWED AS LONG AS THE ASSOCIATED
C RADIAL FUNCTION IS NOT INCLUDED IN THE SET OF FUNCTIONS BEING
C MADE SELF-CONSISTENT.
C
5 5 0 0
0 0 0 0
NA+ 1S 11. 190 3 3 1
2S2/2P6
1S 1 0 0.0 1 0 0 2
2S 2 0 2.0 1 0 0 2
2P 2 1 7.0 1 0 0 6
F 10 10
*
NA 2S 11. 190 4 1 1
2S2/2P6/3S
1S 1 0 0.0 1 0 1 2
2S 2 0 2.0 1 0 1 2
2P 2 1 7.0 1 0 1 6
3S 3 0 10.0 1 0 0 1
F 10 2
*
NA 2P 11. 190 5 1 1
2S2/2P6/3P
1S 1 0 0.0 1 0 1 2
2S 2 0 2.0 1 0 1 2
2P 2 1 7.0 1 0 1 6
3S 3 0 10.0 1 0 1 0
3P 3 1 10.0 1 0 0 1
F 10 2
*
C
C ------------------------------------------------------------------
C 2-3 E X A M P L E 3
C ------------------------------------------------------------------
C
C THIS EXAMPLE ILLUSTRATES THE USE OF MULTIPLE ORBITALS OF THE SAME
C TYPE. THIS FEATURE IS RESTRICTED TO CASES WHERE THE INTERACTION
C BETWEEN CONFIGURATIONS RESULTS IN TERMS WITH AT MOST ONE OVERLAP
C INTEGRAL. THESE ARE ALLOWED ONLY FOR RK AND L INTEGRALS AND SO
C OCCASSIONALLY FK OR GK INTEGRALS MUST BE RE-EXPRESSED AS RK
C INTEGRALS.
C
C
5 5 0 0
0 0 0 0
B 2P 5. 180 4 4 2 1 0 1
2S2/2P 1.0
2P*3 0.3
1S 1 0 0.0 1 0 0 2 2
2S 2 0 2.0 1 0 0 2
2P 2 1 4.0 1 0 0 1
2P* 2 1 3.0 3 0 0 0 3
0.240000000F2( 4 2, 4 2)
-0.94280904R1( 2 2 1, 4 4 2)< 3! 4> 1
F 8 1 0 0 .1D-7 .1D-5
*
C
C
C IN THE ABOVE EXAMPLE THE CFGTOL AND SCFTOL PARAMETERS HAVE BEEN
C RESET TO LARGER VALUES THAN THE DEFAULT VALUES AND NO FUNCTIONS
C ARE PUNCHED. THIS APPROACH IS USEFUL WHEN A TRIAL CALCULATION
C IS BEING PERFORMED.
C
C IN MORE COMPLEX CASES, MORE RESTRICTIONS ARE PRESENT. NORMALLY
C AN OFF-DIAGONAL ENERGY PARAMETER CAN BE COMPUTED DIRECTLY FROM
C A PAIR OF RADIAL EQUATIONS (SEE SEC. 7-3), BUT WHEN NON-ORTHOG-
C ONAL ORBITALS ARE PRESENT, INDICATED BY THE FACT THAT THEIR NL
C VALUES ARE THE SAME, IT MAY BE NECESSARY TO SOLVE A SYSTEM OF
C EQUATIONS FOR SETS OF PARAMETERS. WHEN ORTHO = .FALSE., THE
C PROGRAM ASSUMES OFF-DIAGONAL PARAMETERS CAN BE COMPUTED FROM A
C SINGLE PAIR OF EQUATIONS. WHEN ORTHO = .TRUE., THE PROGRAM
C CHECKS IF OTHER ORBITALS ARE PRESENT, NON-ORTHOGONAL TO EITHER
C ONE OF THE PAIR. AT MOST ONE SUCH ORBITAL FOR EACH MEMBER CAN
C BE ACCOMMODATED. THE IMPLICATION OF THESE RESTRICTIONS CAN BE
C DESCRIBED FOR THE MCHF CALCULATION --
C 2P6.(3S.3P + 3P'.3D )
C IF 2P IS DETERMINED VARIATIONALLY IT MUST BE ORTHOGONAL TO BOTH
C 3P AND 3P' WHICH ARE NON-ORTHOGONAL. HENCE A SYSTEM OF EQUATIONS
C MUST BE SOLVED AND WE NEED ORTHO = .TRUE. ON THE OTHER HAND, IF
C 2P WERE PART OF THE FIXED CORE, THE VALUE OF ORTHO WOULD NOT BE
C IMPORTANT. THE CALCULATION FOR
C 2P6.(3S.3P + 4S.4P + 3P'.3D + 4P'.4D )
C CAN ONLY BE PERFORMED CORRECTLY WITH 2P PART OF A FIXED CORE.
C ORTHO MUST BE .FALSE. OTHERWISE ALL FUNCTIONS WITH DIFFERENT NL
C VALUES WOULD BE MADE ORTHOGONAL. THE -1 OPTION MUST BE USED TO
C INCLUDE 4S,4P IN CONFIGURATION 1, 4P',4D IN CONFIGURATION 3
C FOR ORTHOGONALITY PURPOSES.
C
C
C ------------------------------------------------------------------
C 2-4 E X A M P L E 4
C -------------------------------------------------------------------
C
C THE EXAMPLE BELOW IS A TWO CONFIGURATION PROBLEM WHERE THE INTER-
C ACTION IS SMALL AND ALL FUNCTIONS MUST BE ORTHOGONAL. IN SUCH A
C CALCULATION IT IS IMPORTANT THAT THE FUNCTIONS DEFINING THE DOM-
C INANT CONFIGURATION BE FAIRLY ACCURATE AND SO A ONE CONFIGURATION
C CALCULATION IS USED FOR DETERMINING THE INITIAL ESTIMATES.
C
C THE INTERACTION IN THIS CASE IS SMALL RESULTING IN A SHALLOW
C ENERGY MINIMUM. THIS COMBINED WITH THE MANY ORTHOGONALITY CON-
C STRAINTS MAKES CONVERGENCE DIFFICULT. THE PROGRAM PRINTS OUT
C SEVERAL DIRE MESSAGES BUT THESE DISAPPEAR AS CONVERGENCE SETS IN.
C
5 5 0 0
0 0 0 0
HE 3S 2. 180 2 2 1 0 1 0
1S/2S
1S 1 0 0.0 1 0 0 1
2S 2 0 1.0 1 0 0 1
-.500000000G0( 1 1, 2 1)
F 12 -1
*
HE 3S 2. 180 4 2 2 0 2 2 0 T T
1S/2S 0.9965624
3S/4S -.0016
1S 1 0 0.0 1 0 1 1
2S 2 0 1.0 1 0 1 1
3S 3 0 -5.0 3 0 0 0 1
4S 4 0 -8.7 3 0 0 0 1
-.500000000G0( 1 1, 2 1)
-.5 G0( 3 2, 4 2)
2.0 R0( 1 2 1, 3 4 2)
-2.0 R0( 1 2 1, 4 3 2)
F 15 1 0 0 .1D-7 .1D-6
*
C
C NOTE THAT WHEN NO OVERLAP INTEGRALS ARE PRESENT, THEY CAN SIMPLY
C BE OMITTED. ALSO, NEGATIVE SCREENING PARAMETERS MAY BE USED TO
C TO CONTRACT THE VIRTUAL ORBITALS.
C
C
C ------------------------------------------------------------------
C 2-5 E X A M P L E 5
C -----------------------------------------------------------------
C
C ONE OF THE MORE DIFFICULT CASES, HISTORICALLY WAS 1S2S 1S OF HE.
C THIS IS AN EXAMPLE WHERE LARGE OFF-DIAGONAL ENERGY PARAMETERS
C OCCUR BECAUSE OF THE ORTHOGONALITY CONSTRAINT, THE EFFECT OF
C WHICH IS TO ROTATE THE USUAL 1S, 2S ORBITAL BASIS. ON THE
C OTHER HAND, THE 1S2S 3S CALCULATION IS A PARTICULARLY SIMPLE
C ONE. THE DATA BELOW DESCRIBES A CALCULATION IN WHICH THE
C OUTPUT FOR 3S DEFINES THE INITIAL ESTIMATES FOR THE 1S STATE.
C
C IN THIS CASE, THE 1S STATE IS NOT THE LOWEST STATE OF A GIVEN
C SYMMETRY AND THE HARTREE-FOCK APPROXIMATION IS NOT A PARTICULARLY
C GOOD ONE, THE TOTAL ENERGY BEING CONSIDERABLY BELOW THE OBSERVED.
C AN MCHF APPROXIMATION WHICH INCLUDES 1S2 1S YIELDS A
C MUCH BETTER RESULT. SUCH A CALCULATION FOLLOWS THE SINGLE
C CONFIGURATION CASE BELOW, WITH 3S FUNCTIONS AGAIN SERVING
C AS INITIAL ESTIMATES.
C
C
5 5 0 0
0 0 0 0
HE 3S 2. 180 2 2 1 0 1 0
1S/2S
1S 1 0 0.0 1 0 0 1
2S 2 0 1.0 1 0 0 1
-.500000000G0( 1 1, 2 1)
F 12 -1
*
HE 1S 2. 180 2 2 1 0 1 0
1S/2S
1S 1 0 0.0 1 0 1 1
2S 2 0 1.0 1 0 1 1
1.500000000G0( 1 1, 2 1)
F 12 -1
*
HE 3S 2. 180 2 2 1 0 1 0
1S/2S
1S 1 0 0.0 1 0 1 1
2S 2 0 1.0 1 0 1 1
-.500000000G0( 1 1, 2 1)
F 12 -1
*
HE 1S 2. 180 2 2 2 0 1 1 1 F T
1S/2S 0.9965624
1S2 .114
1S 1 0 0.0 1 0 1 1 2
2S 2 0 1.0 1 0 1 1
1.500000000G0( 1 1, 2 1)
2.82842712 R0( 1 2 1, 1 1 2)
-1.41421356L( 2 1, 1 2)
F 9 -1 .3 0 1.D-7 .1D-6 T
*
C
C
C IN THIS CASE, THE CONFIGURATIONS DIFFER BY EXACTLY ONE ELECTRON
C AND THE INTERACTION NOW ALSO INVOLVES AN L INTEGRAL. THE
C CALCULATIONS HAVE BEEN SET UP IN SUCH A WAY THAT FIRST 1S, THEN
C 2S ARE IMPROVED (IN THE MCHF CASE, 2S IS KEPT STRICTLY
C ORTHOGONAL TO 1S), AFTER WHICH THE FUNCTIONS ARE ROTATED TO
C SATISFY THE STATIONARY CONDITION, AND THE ENERGY MATRIX
C DIAGONALIZED. THE ROTATIONS SEEM TO INTRODUCE OSCILLATIONS
C IN THIS CASE AND SO ACFG WAS SET TO 0.3 IN ORDER TO DAMP OUT
C THESE OSCILLATIONS.
C
C
C ------------------------------------------------------------------
C 2-6 E X A M P L E 6
C -----------------------------------------------------------------
C
C THIS EXAMPLE SHOWS HOW A SIXTH CONFIGURATION CAN BE ADDED TO AN
C INTERACTION MATRIX, WHERE THE MATRIX WAS OUTPUT DURING A PREVIOUS
C CALCULATION. IT IS ASSUMED THAT RADIAL FUNCTIONS CAN BE READ
C FROM UNIT 8.
C
5 5 8 5
9 0 9 9
AR+2 3S 18. 200 6 0 6 2 3 6 0 F F 1
2P5/3P5 0.9920873
2S/3S 0.0364112
3S/3P5/3D(1P) 0.1149757
3S/3P5/3D(3P) -0.0348588
3S/3P5/3D(3P) -0.0015988
2S/3P4/3D -0.018
1S 1 0 0.0 1 0 -1 2 2 2 2 2 2
2S 2 0 3.0 1 0 -1 2 1 2 2 2 1
2P 2 1 7.0 1 0 -1 5 6 5 5 5 6
3S 3 0 10.0 1 0 -1 2 1 1 1 1 2
3P 3 1 10.0 1 0 -1 5 6 5 5 5 4
3D 3 2 12.0 3 0 -1 0 0 1 1 1 1
0.12000000F2( 506, 506)
-0.40000000F2( 506, 606)
0.53333333G1( 506, 606)
-0.08571429G3( 506, 606)
-0.10000000G2( 606, 206)
0.94280904R1( 6 3 6, 5 2 1)
-1.63299316R1( 4 6 6, 5 5 2)
-0.94280904R1( 4 3 6, 5 2 3)
1.41421356R0( 4 3 6, 2 5 3)
-0.81649658R0( 4 3 6, 2 5 4)
-2.30940108R0( 4 3 6, 2 5 5)
F 0 0 0 0 1.D-8 .1D-5
-516.5748431
-0.1543388 -513.0822716
-0.2328091 0.1413721 -514.7085685
0.1012540 -0.2399579 -0.2016125 -514.6445872
0.0062108 -0.0069378 -0.0322798 -0.0086021 -515.0489374
*
C
C NOTICE THAT THE DATA CARDS FOR THE ENERGY EXPRESSION NOW NEED
C REFER ONLY TO THE NEW CONFIGURATION, ALL OTHER ENTRIES OF THE
C INTERACTION MATRIX BEING KEPT FIXED. THIS CAN BE USED TO
C ADVANTAGE IN VERY LARGE PROBLEMS WHERE THE NUMBER OF FK, GK, OR
C RK INTEGRALS MIGHT EXCEED THE MAXIMUM ALLOWED BY THE DIMENSIONS
C OF THE ARRAYS. IN THIS WAY ONE CAN, IN EFFECT, EVALUATE PORTIONS
C OF THE MATRIX AT A TIME.
C
C
C ------------------------------------------------------------------
C 2-7 P E R F O R M A N C E T E S T D A T A
C ------------------------------------------------------------------
C
C
C AN INDICATION OF THE ACCURACY OF THE NUMERICAL PROCEDURE USED IN
C THIS PROGRAM IS GIVEN IN SEC. 6-9 AND ALSO COMPUT. PHYS. COMMUN.
C 4, 107 (1972). THESE RESULTS ALL PERTAIN TO THE HYDROGENIC
C PROBLEM. ADDITIONAL SOURCES OF ERRORS ENTER INTO THE GENERAL HF
C PROBLEM. THE RESULTS FROM THE TEST DATA PROVIDED BELOW, INDICATE
C THE LIMITS OF ACCURACY ON THE IBM 360/370 SERIES, AND ALSO PROVIDE
C INFORMATION ABOUT THE RATE OF CONVERGENCE FOR OUR METHODS. THE
C CASES SELECTED ARE TYPICAL TEST CASES OFTEN USED FOR EVALUATING
C A METHOD.
C
C
5 5 0 0
0 0 0 0
HE 1S 2. 150 1 1 1
1S2
1S 1 0 0.0 1 0 0 2
F 10 10 0 0 .1D-9 .1D-9
*
HE 1P 2. 180 2 2 1 0 1
1S/2P
1S 1 0 0.0 1 0 0 1
2P 2 1 1.0 1 0 0 1
0.50000000G1( 1 1, 2 1)
F 10 10 0 0.1D-12.1D-12
*
HE 3S 2. 180 2 2 1 0 1 0
1S/2S
1S 1 0 0.0 1 0 0 1
2S 2 0 1.0 1 0 0 1
-.500000000G0( 1 1, 2 1)
F 12 -1 0 0 .1D-9 .1D-9
*
HE 1S 2. 180 2 2 1 0 1 0
1S/2S
1S 1 0 0.0 1 0 1 1
2S 2 0 1.0 1 0 1 1
1.500000000G0( 1 1, 2 1)
F 12 -1 0 0 .1D-9 .1D-9
*
LI 4S 3. 200 3 3 1 0 3 0
1S/2S/3S
1S 1 0 0.0 1 0 0 1
2S 2 0 1.0 1 0 0 1
3S 3 0 2.0 1 0 0 1
-.500000000G0( 1 1, 2 1)
-.500000000G0( 1 1, 3 1)
-.500000000G0( 2 1, 3 1)
F 12 -1 0 0 .1D-9 .1D-9
*
W+64 1S 74. 200 3 3 1
1S2/2S2/2P6
1S 1 0 0.0 1 0 0 2
2S 2 0 2.0 1 0 0 2
2P 2 1 7.0 1 0 0 6
F 12 12 0 0.1D-11.1D-11
*
C
C
C THE FOLLOWING TABLE SUMMARIZES THE RESULTS.
C
C T A B L E OF R E S U L T S
C
C -E - TOTAL ENERGY IN ATOMIC UNITS
C PE/KE - RATIO OF POTENTIAL AND KINETIC ENERGY
C DPM - MAXIMUM DIFFERENCE BETWEEN THE FUNCTIONS OF THE
C PREVIOUS AND CURRENT SCF ITERATION
C NXCH - NUMBER OF EXCHANGE FUNCTIONS COMPUTED
C
C
C -E (PE/KE)+2 DPM NXCH
C 1. HE 1S2 1S 2.861679995 4.D-9 1.38D-10 13( 8)
C 2. HE 1S/2P 1P 2.122464215 2.D-9 2.33D-14 19( 7)
C 3. HE 1S/2S 3S 2.174250777 -12.D-9 1.49D-08 18(12)
C 4. HE 1S/2S 1S 2.169854456 -1.D-9 7.37D-09 24(20)
C 5. LI 1S/2S/3S 4S 5.204454130 -16.D-9 4.29D-09 27(21)
C 6. W+64 1S2/2S2/2P6 1S 10318.516188029 1.D-9 2.00D-12 24(12)
C
C THE METHODS USED IN THE PROGRAM RELY ON PROPERTIES OF THE HF
C EQUATIONS, PARTICULARLY WHEN SATISFYING THE ORTHOGONALITY REQUIR-
C EMENT. THE DISCRETIZED APPROXIMATION WILL HAVE THESE PROPERTIES
C ONLY TO LIMITED ACCURACY. FOR EXAMPLE, THE DISCRETIZED PROBLEM
C FOR HE 1S/2S 3S WITH OFF-DIAGONAL ENERGY PARAMTETERS EQUAL TO
C ZERO, HAS SOLUTIONS FOR WHICH <1S!2S> = 1.4D-8. THE ORTHOGON-
C IZATION PROCESS THEN LIMITS THE APPARENT SCF CONVERGENCE. HOW-
C EVER, A SMALLER DPM DOES NOT NECESSARILY QUARANTEE GREATER ACC-
C URACY. A BETTER INDICATION OF ACCURACY IS THE RATIO, (PE/KE)+2,
C WHICH FOR EXACT SOLUTIONS WOULD BE ZERO. IN MOST CASES THE
C MAGNITUDE OF THIS QUANTITY IS SEVERAL UNITS IN THE NINTH DECIMAL
C PLACE.
C
C AN INDICATION OF THE EFFECTIVENESS OF THE PROGRAM CAN BE OBTAINED
C FROM THE NUMBER OF TIMES AN EXCHANGE FUNCTION HAS TO BE COMPUTED.
C (IN LARGE ATOMS MOST OF THE COMPUTATION TIME IS DEVOTED TO THE
C EVALUATION OF EXCHANGE FUNCTIONS, AND SO THIS IS A REASONABLY
C CONVENIENT MACHINE INDEPENDENT MEASURE OF EFFICIENCY, WHERE THE
C MOST EFFICIENT PROGRAM ACHIEVES A GIVEN ACCURACY WITH THE
C LEAST AMOUNT OF COMPUTATION.) THE FIRST NUMBER UNDER THE NXCH
C COLUMN IS THE ACTUAL NUMBER OF CALLS TO XCH MADE BY THE PROGRAM,
C BUT BECAUSE OF THE HIGH ACCURACY REQUESTED, SEVERAL OF THESE DID
C NOT IMPROVE THE DEGREE OF SELF-CONSISTENCY. IN PARENTHESES IS
C THE NUMBER OF CALLS TO XCH REQUIRED FOR AN A POSTERIORI SELF-
C CONSISTENCY OF AT LEAST 8.5D-8, WHERE THE DEGREE OF SELF-CONSIST-
C ENCY IS NOW THE MAXIMUM DIFFERENCE BETWEEN THE CURRENT FUNCTION
C AND ITS NEXT SCF ITERATE. NOTICE THE VERY RAPID RATE OF CONVER-
C GENCE FOR HIGHLY IONIZED SYSTEMS. FOR W+64, AN AVERAGE OF 4.33
C IMPROVEMENTS PER FUNCTION PRODUCED AN A POSTERIORI DEGREE OF
C SELF-CONSISTENCY OF AT LEAST 6.6D-9 . A MORE DETAILED STUDY OF
C THE RATE OF CONVERGENCE FOR THE VARIOUS CASES WILL APPEAR IN THE
C JOURNAL OF COMPUTATIONAL PHYSICS, 1977.
C
C
▶EOF◀