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Semi-Empirical Methods

There are a variety of semi-empirical methods available in Gaussian 16. The AM1 and the PM3 methods have been reimplemented [Frisch09, Thiel96, Thiel92] to use the standard integral processing infrastructure (rather than using code from the public-domain MOPAC). In addition to increased efficiency, this change also provides analytic gradients and frequencies. PM6 and PDDG are also implemented in this way. The remaining semi-empirical methods use the modified version of MOPAC in Link 402, and they are discussed on their individual pages.

No basis set keyword should be specified with any semi-empirical keyword.

Standard parameters for supported atoms are generated automatically by the program unless the NoGenerate option is specified. Additional and/or alternate ones can also be read-in in several ways (see the options). Read-in parameters take precedence over internal ones when both are used.

Using PM7 with Third-Row and Higher Elements

For some systems containing these elements, instabilities can exists in the PM7 wavefunction. There may be difficulties getting the wavefunction to converge. This is less common for lighter elements and is most likely to occur for transition metal complexes. In such cases, the first option is to try an alternate SCF algorithm: e.g., SCF=YQC. If this fails, other options are to use a different initial guess or read in the density from a calculation using a different method (see the Guess keyword).

In some cases, the PM7 wavefunction that is found may not be the lowest energy SCF solution. This too is most common for transition metals and so caution is advised when performing calculations on transition metals. A straightforward way of testing the wavefunction is to use the Stable keyword.


Generate the standard parameters for the specified method. This is the default. NoGenerate says not to generate any standard parameters; all parameters must be read in.


Read parameters from the input stream in Gaussian's format. Any parameter can be specified or overridden. The input section must be terminated by a blank line. Cards is a synonym for Input.


Read parameters from the input stream in MOPAC's external format and units. Most but not all parameters can be changed. The input section must be terminated by a blank line.


Read parameters from the input stream, first in Gaussian's format, followed by more parameters in MOPAC's format. Both input sections must be terminated by a blank line.


Read parameters from the checkpoint file. Chk and Read are synonymous with Checkpoint.


Read parameters from the checkpoint file if present; otherwise generate them.


Read parameters from the read-write file.


Print parameters used for elements in the current job in Gaussian's format. This is the default if parameters are read from the input file. NoPrint says not to print parameters, and it is the default if standard parameters are used.


Print parameters for all elements (in Gaussian's format), even the ones that are not present in the molecule specification.


Print all parameters including the ones that are zero. The default is NonZero, which says to print only non-zero parameters.


Use the old MOPAC-based code. Second derivatives are done numerically. Applies only to AM1 and PM3. The default is New, which says to use the new implementation described above.

Semi-empirical parameters can be specified in two different formats, Gaussian and MOPAC, via the Input and MOPACExternal options (respectively). We begin with the native Gaussian semi-empirical parameter format, which is very general.

Here's an example in Gaussian format, for FeCH (actual output may wrap differently):

 Initial section of global parameters.
 Method=40 CoreType=2 PM6R6=0.0000124488 PM6R12=0.0000007621
 H	Parameters for hydrogen.
 PQN=1 NValence=1 F0ss=0.5309794634 ZetaOverlap=1.2686410000 U=-0.4133181193
 Beta=-0.3069665271 CoreKO=0.9416560046 KON=0,0,0,0.9416560046 EISol=-0.4133181193
 GCore=0.0016794859,0.8557539899,3.3750716603 DCore=1,3,1.8737858033,2.2435870000
 C	Parameters for carbon.
 PQN=2,2 NValence=4 F0ss=0.4900713271 F0sp=0.4236511476 F0pp=0.3644399975 F2pp=0.1978513243
 Beta=-0.5653970441,-0.2745883502 DDN=0,1,0.7535642510 DDN=1,1,0.7192361890
 CoreKO=1.0202596487 KON=0,0,0,1.0202596487 KON=1,0,1,1.2918442312 KON=0,1,1,1.0202596487
 KON=2,1,1,0.7626764584 EISol=-4.2335803497 EHeat=0.2723305520 DipHyp=1.5070417957
 GCore=0.0032154961,0.5881175739,2.5208171825 DCore=1,4,0.2878149911,0.2165060000
 DCore=2,3,1.6101301385,3.2139710000 DCore=3,3,1.7155258339,16.1800020000 DCore=4,3,2.2293611369,25.0358790000
 DCore=5,3,1.5446719761,1.8748590000 DCore=6,5,1.3831173494,0.8135100000,3.1644797074,9.2800000000
 Fe	Parameters for iron.
 PQN=4,4,3 NValence=8 F0ss=0.2931506917 F0sp=0.2861621092 F0pp=0.2797829041
 F0sd=0.3417747898 F0pd=0.3378189937 F0dd=0.5580709105 F2pp=0.1567537881 F2pd=0.1236661383
 F2dd=0.2945882511 F4dd=0.1921227725 G1sp=0.2072870321 G1pd=0.1102204721 G2sd=0.0588483485
 G3pd=0.0671224585 Rsppd=0.1364112343 Rsdpp=0.1031169651 Rsddd=0.1510228569
 Beta=0.2950096563,-0.0413709206,-0.1288993981 DDN=0,1,0.0896587028 DDN=1,1,0.3534210933
 DDN=0,2,1.6776352014 DDN=1,2,0.0796789968 DDN=2,2,1.3085519205 CoreKO=1.2720920000
 KON=0,0,0,1.7056074374 KON=1,0,1,0.2359511557 KON=0,1,1,1.7056074374 KON=2,1,1,0.4977547907
 KON=2,0,2,1.6958541322 KON=1,1,2,0.2947417183 KON=0,2,2,0.8959434914 KON=2,2,2,1.2449263774
 EISol=-15.6859079709 EHeat=0.1582446241 DipHyp=0.1793070893 DCore=1,3,0.4521755746,0.0251950000
 DCore=6,3,2.1121277473,0.3668350000 DCore=7,3,1.3232002016,0.1553420000 DCore=8,3,0.9135254945,0.1364220000
 DCore=9,3,2.2726610620,3.6573500000 DCore=15,3,1.3586804751,0.4312910000 DCore=16,3,0.5233514967,0.0334780000
 DCore=17,3,0.6507784269,0.0194730000 DCore=19,3,1.0583544172,6.0000000000

Atomic units are used throughout the input. Parameter sections are separated by lines containing four asterisks.

The following items are specified in the global section:

Method    An integer corresponding to the desired semi-empirical method. This value should correspond to the method specified in the route section as a check. The values are 8 for AM1, 9 for PM3, 10 for PM3MM, 40 for PM6 and 41 for PDDG.
CoreType    Type of core repulsion terms, where 1 means AM1, PM3, or PDDG, and 2 means PM6.
PeptideFC    Force constant for peptide linkages. Only valid for with PM3MM.
RIJScale    Rij scale factor for the AM1 O-H and N-H bonds.
PM6R6    R6 parameter for the PM6 core repulsion.
PM6R12    R12 parameter for the PM6 core repulsion.

The following items specify the parameters for an element:

PQN    Principal quantum numbers for each shell (s, p, d). Determines which basis functions are used on the element.
NValence    Number of valence electrons.
ZetaOverlap    Slater exponents for basis functions used in the calculation of the overlap contribution to the core Hamiltonian.
Zeta1C    Slater exponents for basis functions used in the computation of those one-center two-electron integrals that were not specified explicitly.
F0*, G*, Rs*    Slater-Condon parameters for one-center two-electron integrals. If any of these items are not specified, they are computed from the Zeta1C exponents. When internal parameters are printed, all values are included regardless of whether they were computed from Zeta1C or a specific value. The full list of these parameters is: F0ss, F0sp, F0pp, F0sd, F0pd, F0dd, F2pp, F2pd, F2dd, F4dd, G1sp, G1pd, G2sd, G3pd, Rsppd, Rsdpp and Rsddd.
U    Diagonal core Hamiltonian matrix elements, one per angular momentum.
Beta    Off-diagonal core Hamiltonian parameters, one per angular momentum.
DDN    Point-charge distance parameters for multipole-approximated two-center two-electron integrals. Each instance has the form L1,L2,Value and applies to charge distributions involving one basis function of angular momentum L1 and one of angular momentum L2. If any are needed but are not specified, they are computed from the Zeta1C exponents.
KON    Klopman-Ohno parameters for two-center two-electron integrals. Required but unspecified items are computed by matching the one-center limit to the one-center integrals given by the Slater-Condon parameters and Zeta1C, and using the specified or defaulted DD values. Each instance is of the form LT,L1,L2,Value and applies to the LT angular momentum component of the product of functions of angular momentum L1 and L2.
CoreKO    Klopman-Ohno parameter used in nuclear attraction terms. If not specified, the 0,0,0 (L=0 SS) parameter is used.
EHeat    Heat of formation of the isolated atom.
EISol    Energy of the isolated atom. If not provided, it is computed from the other parameters and a standard electronic configuration for the atom.
DipHyp    Dipole moment hybridization parameter.
DCore    Core repulsion parameters. Each instance is of the form El,IType,Value1,Value2. Each term specifies the core repulsion between the current element and the element El. IType specifies the bond type: 1 for usual AM1, 2 for AM1 N-H and O-H, 3 usual PM6, 4 PM6 O-H, 5 PM6 CC triple bond, and 6 PM6 Si-O. There will be one or two parameters values, depending on the specific functional form.

MOPAC-style semi-empirical parameter input. If you request PM6=MOPACExternal or AM1=MOPACExternal, then an input section is read giving parameters in the same formation used by the External keyword in MOPAC. This is less general than the native Gaussian format, but includes the most common parameters. Refer to the MOPAC documentation for details. The units expected by Gaussian in this format are those expected by MOPAC, which are a mixture of atomic and other units.

The following table gives the correspondence between MOPAC External labels and the native Gaussian input:

MOPAC Gaussian
ZS,ZP,ZD ZetaOverlap
BetaS,BetaP,BetaD Beta
GSS,GPP,…,FODD,… F0ss,F0pp, etc.†
DD2 DDN=0,1,Value
DD3 DDN=1,1,Value
DD4 DDN=0,2,Value
DD5 DDN=1,2,Value
DD6 DDN=2,2,Value
PO1 KON=0,0,0,Value
PO2 KON=1,0,1,Value
PO3 KON=2,1,1,Value
PO4 KON=2,0,2,Value
PO5 KON=1,1,2,Value
PO6 KON=2,2,2,Value
PO7 KON=0,1,1,Value
PO8 KON=0,2,2,Value
PO9 CoreKO
EHeat EHeat
AlpB_NN,XFac_NN DCore=NN,3,Alpha,XFac

†Note that MOPAC’s GSP and GP2 are linear combinations of F0sp and G1sp. Gaussian uses the standard Slater-Condon names and parameter definitions.

Here is example MOPAC External data for Cr, printed by MOPAC using its debug option:


  NI  TYPE        VALUE     UNIT
  24  USS     -34.86433900  EV        ONE-CENTER ENERGY FOR S
  24  UPP     -26.97861500  EV        ONE-CENTER ENERGY FOR P
  24  UDD     -54.43103600  EV        ONE-CENTER ENERGY FOR D
  24  ZS        3.28346000  AU        ORBITAL EXPONENT  FOR S
  24  ZP        1.02939400  AU        ORBITAL EXPONENT  FOR P
  24  ZD        1.62311900  AU        ORBITAL EXPONENT  FOR D
  24  BETAS    -5.12261500  EV        BETA PARAMETER    FOR S
  24  BETAP     3.92671100  EV        BETA PARAMETER    FOR P
  24  BETAD    -4.23055000  EV        BETA PARAMETER    FOR D
  24  GSS       8.85557242  EV        ONE-CENTER INTEGRAL (SS,SS)
  24  GPP       5.05309383  EV        ONE-CENTER INTEGRAL (PP,PP)
  24  GSP       5.58863066  EV        ONE-CENTER INTEGRAL (SS,PP)
  24  GP2       4.42952965  EV        ONE-CENTER INTEGRAL (PP*,PP*)
  24  HSP       0.64803936  EV        ONE-CENTER INTEGRAL (SP,SP)
  24  ZSN       1.61985300  AU        INTERNAL EXPONENT FOR S - (IJ,KL)
  24  ZPN       0.84826600  AU        INTERNAL EXPONENT FOR P - (IJ,KL)
  24  ZDN       1.40501500  AU        INTERNAL EXPONENT FOR D - (IJ,KL)
  24  F0DD      9.86923654  EV        SLATER-CONDON PARAMETER F0DD
  24  F2DD      5.20966257  EV        SLATER-CONDON PARAMETER F2DD
  24  F4DD      3.39760602  EV        SLATER-CONDON PARAMETER F4DD
  24  F0SD      6.15013600  EV        SLATER-CONDON PARAMETER F0SD
  24  G2SD      2.00030000  EV        SLATER-CONDON PARAMETER G2SD
  24  F0PD      5.63536196  EV        SLATER-CONDON PARAMETER F0PD
  24  F2PD      1.91648791  EV        SLATER-CONDON PARAMETER F2PD
  24  G1PD      1.58022558  EV        SLATER-CONDON PARAMETER G1PD
  24  G3PD      0.96233144  EV        SLATER-CONDON PARAMETER G3PD
  24  DD2       0.28669123  BOHR      CHARGE SEPARATION, SP, L=1
  24  DD3       2.91433601  BOHR      CHARGE SEPARATION, PP, L=2
  24  DD4       1.12394737  BOHR      CHARGE SEPARATION, SD, L=2
  24  DD5       0.81804068  BOHR      CHARGE SEPARATION, PD, L=1
  24  DD6       1.23219554  BOHR      CHARGE SEPARATION, DD, L=2
  24  PO1       1.53639890  BOHR      KLOPMAN-OHNO TERM, SS, L=0
  24  PO2       0.72875078  BOHR      KLOPMAN-OHNO TERM, SP, L=1
  24  PO3       1.96024483  BOHR      KLOPMAN-OHNO TERM, PP, L=2
  24  PO4       0.94950312  BOHR      KLOPMAN-OHNO TERM, SD, L=2
  24  PO5       1.66105265  BOHR      KLOPMAN-OHNO TERM, PD, L=1
  24  PO6       1.08400979  BOHR      KLOPMAN-OHNO TERM, DD, L=2
  24  PO7       1.53639890  BOHR      KLOPMAN-OHNO TERM, PP, L=0
  24  PO8       1.37859617  BOHR      KLOPMAN-OHNO TERM, DD, L=0
  24  PO9       1.53639890  BOHR      KLOPMAN-OHNO TERM, CORE
  24  CORE      6.00000000  E         CORE CHARGE
  24  EISOL  -185.72482255  EV        TOTAL ENERGY OF THE ATOM (CALC)
  24 ALPB_24   4.65541900  ALPB factor
  24 XFAC_24  10.31860700  XFAC factor

Energies, optimizations and frequencies.

Parameters are stored in the program for the following elements:

  • AM1: H, Li-F, Mg-Cl, Cr, Zn, Ge, Br, Sn, I and Hg.
  • PM3: H, Li-F, Na-Cl, K, Ca, Cr, Zn-Br, Rb, Sr, Cd-I, Cs, Ba and Hg-Bi.
  • PM6: H-Ba and Lu-Bi.
  • PM7: H-La and Lu-Bi.

The energy from these calculations appears in the output file as follows:

 SCF Done: E(RAM1) = -0.943839275843E-01 A.U. after 6 cycles AM1
 SCF Done: E(RPM3) = -0.850201514485E-01 A.U. after 6 cycles PM3
 SCF Done: E(RPM6) = -0.864068239687E-01 A.U. after 7 cycles PM6
 SCF Done: E(RPM7) = -0.919784216493E-01 A.U. after 7 cycles PM7

The energy printed is the heat of formation as computed by the model. Energies for other semi-empirical methods are reported similarly.

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