Subject: Re: Calculations without Born-Oppenheimer approximation "2nd Call"

From: "Dr. N. SUKUMAR" <nagams(at)rpi.edu>
Date: Wed, 17 Dec 2003 13:29:14 EST

On Wed, 17 Dec 2003 09:40:09 -0500 "Shobe, David" wrote:

> That brings up an interesting question (interesting for me anyway).  What

> Dr. Sukumar defined below is AFAIK neither r-sub-e as it's usually
> defined within the Born-Oppenheimer approximation (i.e. the internuclear
> distance with minimum electronic energy), nor r-sub-zero which is the
> *mean* of the the two-particle radial distribution function.	The
> definition below is the *mode* of the distribution.  
> 

The mean of the two-particle radial distribution function is not a
meaningful quantity. Note that when you solve for the eigenfunctions of the
Hamilronian WITHOUT making the Born-Oppenheimer separation, you do NOT get
solutions corresponding to a particular molecular structure or isomer; you
get a wave function which corresponds to ALL isomers of that stoichiometry.
Thus the two-particle distribution function for C6H6 will have peaks
corresponding to benzene, cyclobutadiene, prismane, bicyclo-propenyl, etc.
The mean of these structures makes no sense chemically.

In making the correspondance between the peaks (mode) of the two-particle 
radial distribution functions and equilibrium bond lengths, one makes the
assumption that the low-energy regions of the potential energy
hypersurfaces will be more populated in a canonical ensemble and thus will
correspond to maxima in the distribution function. This follows equally
> from Boltzman or Fermi-Dirac/Bose Einstein statistics for the nuclei. These
would then correspond to low-temperature experimental structures. For
high-temperature structures or highly fluxional molecules, of course, this
simple correspondance with experiment will not hold.

N. Sukumar
RPI Department of Chemistry

> 
> ++++++++++++++++++++++++++++
> Dr. N. SUKUMAR wrote:
> "equilibrium bond lengths, for instance,would be defined as the peaks in
> the two-particle radial distribution functions for the two bonded nuclei"

---

From: "Shobe, David" <dshobe)at(sud-chemieinc.com>
Date: Wed, 17 Dec 2003 09:40:09 -0500

That brings up an interesting question (interesting for me anyway).  What Dr. Sukumar defined below is AFAIK neither r-sub-e as it's usually defined within the Born-Oppenheimer approximation (i.e. the internuclear distance with minimum electronic energy), nor r-sub-zero which is the *mean* of the the two-particle radial distribution function.  The definition below is the *mode* of the distribution.  

Dr. N. SUKUMAR wrote:
"equilibrium bond lengths, for instance,would be defined as the peaks in the two-particle radial distribution functions for the two bonded nuclei"

There are experimentally determined r-sub-e's, for example in the NIST Webbook databases. Is this the definition of r-sub-e used by experimentalists?  It wouldn't necessarily be equal to r-sub-e as defined by computational chemists using the Born-Oppenheimer approximation, but one could argue that this is the result of using an approximation, and not that there are two different parameters called r-sub-e.

--David Shobe

++++++++++++++++++++++++++++
Dr. N. SUKUMAR wrote:
"equilibrium bond lengths, for instance,would be defined as the peaks in the two-particle radial distribution functions for the two bonded nuclei"

-----Original Message-----
From: Computational Chemistry List [mailto:chemistry-request)at(ccl.net]On
Behalf Of Dr. N. SUKUMAR
Sent: Tuesday, December 16, 2003 6:03 PM
To: gkalathi)at(vub.ac.be
Cc: chemistry)at(ccl.net
Subject: CCL:Calculations without Born-Oppenheimer approximation "2nd
Call"

On Tue, 16 Dec 2003 15:11:47 +0100 (MET) Giju Kalathingal wrote:

> Last week I had send the following message without 
> any success. So I am posting it again.
> 
> Dear CCL-all,
> 
> Could you please give me some info on programs
> that let me to do quantum chemical calculations
> (geometry optimzations and vibrational frequencies) 
> without Born-Oppenheimer approximation?
> 
> Thanks in advance.
> 
> Giju Kalathingal
> 
> ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 
> 

Perhaps the reason you did not get any response was because your question
specifically dealt with programs for geometry optimzations and vibrational
frequencies. These concepts (molecular geometry and molecular vibrations)
are normally understood within the Born-Oppenheimer approximation. I do not
know of any readily available programs that deal with these outside of BO.
However quantities such as bond lengths and vibrational frequencies are
derivable from experiment and thus these concepts ARE defined beyond the
Born-Oppenheimer approximation: equilibrium bond lengths, for instance,
would be defined as the peaks in the two-particle radial distribution
functions for the two bonded nuclei. See the special issue of Israel
Journal of Chemistry, vol.19 (1980) for many papers (and much debate), e.g.
those of R.G.Woolley, pp.30-46 and of Carl Trindle, pp.47-53. L. Lathouwers
and P. Van Leuven (CPL 52, 439, 1977; PRA 18, 2150, 1978; IJQC S12, 371,
1978) demonstrated that the generator coordinate method could be employed
to avoid the BO separation and develop a non-adiabatic formulation of
molecular quantum theory.

In a series of pioneering papers on hydrides (CPL 3, 705, 1969; PS 185, 90,
1969; PRA 2, 728, 1200, 1675, 1970; PRA 3, 565, 1022, 1971; PRA 4, 457,
1971; PRA 5, 1104, 1972), I.L.Thomas put electrons into electronic orbitals
and protons into protonic orbitals (thereby avoiding the BO approximation),
antisymmetrized the two kinds of orbitals separately, performed SCF
computations and obtained molecular structures and spectra that
corresponded to experiment. For instance, the microwave transition
frequency of ammonia (conventionally explained by "umbrella flipping"
between two structures) is now obtained as the transition frequency between
protonic orbitals. Subsequently Thomas went on to become vice-president of
Occidental Petroleum and did not continue his protonic structure studies. I
wrote some programs to do similar computations as part of my candidacy work
at Stony Brook in the early 80s, but those are now lost on some magnetic
tape somewhere.

There are now several computations and programs that go beyond the BO
approximation by calculating the nonadiabatic coupling terms or performing
a diabatic transformation in multi-reference CI or CASSCF spaces (see e.g.
D.R.Yarkony in "Modern Electronic Structure Theory" Part I, World
Scientific, Singapore, 1995, pp.642-721; W.Domcke, et al. CPL 216,362,
1993; 226, 257, 1994; Neuheuser, Sukumar and Peyerimhoff, Chem.Phys., 194,
45-64, 1995; Mol.Phys. 95, 61-70, 1998) but none of these can be considered
as truly "without Born-Oppenheimer approximation"

(PRA=Phys.Rev A; CPL=Chem.Phys.Lett.; IJQC=Int.J.Quantum Chem)

N. Sukumar

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