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I am trying to understand the working principle of basis sets in quantum chemistry. As far as I understand in the Hartree-Fock method, the basis set is used to calculate the matrix representation of Fock operator in each iteration in the SCF procedure. Usually each basis function in a chosen basis set is represented as a linear combination of several primitive Gaussian type orbital (GTO). I want to use cc-PVDZ basis for C atom which I can take from the EMSL Basis Set Exchange Library. For example in the Gaussian format:

!  cc-pVDZ  EMSL  Basis Set Exchange Library   10/19/17 2:40 AM
! Elements                             References
! --------                             ----------
! H     : T.H. Dunning, Jr. J. Chem. Phys. 90, 1007 (1989).
! He    : D.E. Woon and T.H. Dunning, Jr. J. Chem. Phys. 100, 2975 (1994).
! Li - Ne: T.H. Dunning, Jr. J. Chem. Phys. 90, 1007 (1989).
! Na - Mg: D.E. Woon and T.H. Dunning, Jr.  (to be published)
! Al - Ar: D.E. Woon and T.H. Dunning, Jr.  J. Chem. Phys. 98, 1358 (1993).
! Sc - Zn: N.B. Balabanov and K.A. Peterson, J. Chem. Phys. 123, 064107 (2005),
! N.B. Balabanov and K.A. Peterson, J. Chem. Phys. 125, 074110 (2006)
! Ca     : J. Koput and K.A. Peterson, J. Phys. Chem. A, 106, 9595 (2002).
! 



****
C     0 
S   8   1.00
   6665.0000000              0.0006920        
   1000.0000000              0.0053290        
    228.0000000              0.0270770        
     64.7100000              0.1017180        
     21.0600000              0.2747400        
      7.4950000              0.4485640        
      2.7970000              0.2850740        
      0.5215000              0.0152040        
S   8   1.00
   6665.0000000             -0.0001460        
   1000.0000000             -0.0011540        
    228.0000000             -0.0057250        
     64.7100000             -0.0233120        
     21.0600000             -0.0639550        
      7.4950000             -0.1499810        
      2.7970000             -0.1272620        
      0.5215000              0.5445290        
S   1   1.00
      0.1596000              1.0000000        
P   3   1.00
      9.4390000              0.0381090        
      2.0020000              0.2094800        
      0.5456000              0.5085570        
P   1   1.00
      0.1517000              1.0000000        
D   1   1.00
      0.5500000              1.0000000        
****
  1. In the lines such as S 8 1.00 I know that the letter S is the orbital type, 8 is the number of GTO in this orbital's expansion, but what is 1.00 for?

  2. The GTO for a p-type orbital has the form $x^l y^m z^n \exp(-ar^2)$ where the powers satisfy $l+m+n=1$ and thus there are three possible sets of $\{l,m,n\}$. But for instance in that link for lines with p-type orbitals, which possibilities shall I pick? The same question for d-type orbitals.

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    $\begingroup$ The link is broken. I have used my crystal ball to guess that you looked at a Gaussian94-style listing. In this case, the 1.0 refers to a sort of global scaling factor. In general, see reference of the program the format of which you use for a description of the basis set listing format. In terms of which p-orbitals to pick: all of them. You never want just one - imagine your molecule rotates. $\endgroup$ – TAR86 Oct 19 '17 at 8:50
  • $\begingroup$ Yes you are right it's Gaussian94 style for cc-PVDZ basis set for C atom. Using example in that link, there are 8 listed orbitals, 3 of them p, one of them d, the rest is s. Does this mean there are in total 4+(3x3)+(1x6)=19 orbitals to be used to represent the Fock operator? $\endgroup$ – nougako Oct 19 '17 at 9:01
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    $\begingroup$ Your calculation is correct for "cartesian" gaussians (noted as 6D in Gaussian03 etc.). Many programs do linear combinations of the cartesian gaussians to yield "spherical" gaussians, that is, 5 d functions and one s function (which I imagine has nodal planes). The s function is then effectively discarded to save computational time. $\endgroup$ – TAR86 Oct 19 '17 at 12:15
  • $\begingroup$ @nougako I think you confused Dunning's cc-pCVDZ basis set with the cc-pVDZ basis set. Martin has posted the cc-pVDZ basis set as you said, but from your comment I assume you were looking at the core polarizied double zeta basis set (which has an additional sp shell). Could you state which basis set your intented to use? $\endgroup$ – awvwgk Nov 9 '17 at 19:01
  • $\begingroup$ I intended to use cc-pVDZ for C atom. It's been a while now and I don't remember why I said there are 8 orbitals there while it seems like there are only 6 in my previous comment. $\endgroup$ – nougako Nov 11 '17 at 2:16
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Have a look in the Gaussian manual if something is unclear.

If you want to use an extern basis set in Gaussian you have to respect the gaussian input conventions for the basis set, which you can find here (gaussian.com/gen/). Since you are using the EMSL Basis Set Exchange the server gives you a file that satisfies these conventions. Like:

Type NGauss Sc
$α_1\quad d_{1μ}$
$α_2\quad d_{2μ}$

$α_N\quad d_{Nμ}$

Type stands for the shell, then you have the number of primitives with NGauss and scaling factor Sc. Afterwards you put your exponents and coeffients of the primitive gaussian functions which then will be contracted to the respective contracted gaussian functions. You might notice that the scaling factor is set to unity for almost all basis sets—I checked some Dunning, Pople and Ahlrichs basis sets and always found it set to unity, which seems resonable. In the TURBOMOLE format the scaling factor is omitted totally, so it seems to be a Gaussian specific hack.

Now to your second question. You are specifing the shell, not a single function, a shell contains always all spherical harmonics belonging to the given quantum number or all cartesian gaussian functions. The spherical harmonics are the solution of the Schrödinger equation for an electron on a surface. You have generally $2l+1$ spherical harmonics where $l$ is the azimudal quantum number specified as 0/s, 1/p, 2/d, 3/f, 4/g, 5/h and so on. There exists also the possibilty to use cartesian gaussians with $(l+1)(l+2)/2$ degenerated functions.

Why am I telling you that? You always choose a whole shell with a azimudal quantum number, how many function belong to this shell depends on the quantum chemistry code you are using. ASAIK, Gaussian supports both.

In case of a p-shell it does not matter, both expressions evaluate to three p-functions (as expected). So you get three p-functions with one p-shell.

The basis set you have chosen would be evaluted as:

(9s4p1d) → [3s2p1d]

Having nine primitive s type gaussians (note that Dunning basis sets use a general contraction scheme), four primitive p type gaussians and one primitive d type gaussian which form a basis set out of three s shells, two p shells and one d shell with 14 contracted spherical harmonic gaussian functions or 15 contracted cartesian gaussian functions for carbon.

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