From owner-chemistry@ccl.net Tue Sep 22 11:58:01 2020 From: "Dr.N Sukumar n.sukumar]=[snu.edu.in" To: CCL Subject: CCL: real-valued versus complex-valued molecular orbitals Message-Id: <-54171-200922044131-19746-1/XyTTL8R8TqjnqI5dMURg[]server.ccl.net> X-Original-From: "Dr.N Sukumar" Content-Type: multipart/alternative; boundary="000000000000a60d0205afe2ee23" Date: Tue, 22 Sep 2020 14:11:12 +0530 MIME-Version: 1.0 Sent to CCL by: "Dr.N Sukumar" [n.sukumar*|*snu.edu.in] --000000000000a60d0205afe2ee23 Content-Type: text/plain; charset="UTF-8" I'm not sure whether I've understood your questions correctly. But here are my responses: 1. The eigenfunctions of a Hermitian operator form a complete set. So if the Hamiltonian for the system is Hermitian, then in the limit of a complete basis (which is, of course, not realized in practice), an appropriate expansion in such a basis (whether real or complex) should give the true ground state. 2. "*that using complex valued orbitals would not reach a lower energy state*" - this is forbidden by the variation theorem, the validity of which does not rely upon the real or complex nature of the basis. 3. "*Are there any kinds of chemical bonds that exist for complex-valued molecular orbitals that do not exist for real-valued molecular orbitals?*" - A linear combination of degenerate eigenfunctions of a Hermitian operator will also be an eigenfunction with the same eigenvalue. So an appropriate linear combination of degenerate complex orbitals can be constructed to give a real orbital with the same energy. For example, if the Hamiltonian is real, psi and psi* will be degenerate, and psi+psi* will give a real orbital with the same energy. 4. The symmetry labels sigma, pi, delta,... are strictly valid only for linear molecules. Since molecular orbitals typically extend over multiple centers, the appropriate symmetry designations to use for the molecular orbitals will be the irreducible representations of the point group of the molecule. *N. SukumarProfessor of ChemistryDirector, Center for Informatics**Shiv Nadar University, India* https://chemistry.snu.edu.in/people/faculty/n-sukumar "The ability to reduce everything to simple fundamental laws does not imply the ability to start from those laws and reconstruct the universe." - Phillip W. Anderson, in "More is Different" (1972) On Tue, Sep 22, 2020 at 11:54 AM Thomas Manz thomasamanz**gmail.com < owner-chemistry() ccl.net> wrote: > Dear colleagues, > > In my research, I've encountered a question about real-valued versus > complex-valued molecular orbitals. Normally, most software programs that > perform DFT, Hartree-Fock, coupled cluster, or configuration interaction > type quantum chemistry computations on molecules use real-valued > molecular orbitals. > > My first question is whether there exists any theorem that shows this will > converge to the ground state (i.e., that using complex valued orbitals > would not reach a lower energy state) when the Hamiltonian does not contain > any spin-orbit coupling or applied magnetic field? In other words, when the > multi-electronic Hamiltonian is comprised of the normal terms: electron > kinetic energy, nuclear-electron potential energy, nuclear-nuclear > potential energy, electron-electron Coulomb & exchange-correlation energies. > > My second question is regarding periodic DFT calculations for which many > software programs use complex-valued molecular orbitals. Are their any > kinds of chemical bonds that exist for complex-valued molecular orbitals > that do not exist for real-valued molecular orbitals? Are their any new > bonding symmetries made possible for the complex-valued orbitals that > cannot exist for the real-valued orbitals? For real-valued molecular > orbitals, the primary covalent bond-orbital symmetries are sigma, pi, > delta, and phi. Do complex-valued orbitals enable any additional > bond-orbital symmetries? > > I sincerely appreciate any insights into this topic you can provide. > > Tom > --000000000000a60d0205afe2ee23 Content-Type: text/html; charset="UTF-8" Content-Transfer-Encoding: quoted-printable
I'm not sure whether I've understood your que= stions correctly. But here are my responses:
  1. The eigenfun= ctions of a Hermitian operator form a complete set. So if the Hamiltonian f= or the system is=20 Hermitian, then in the limit of a complete basis (which is, of course, not = realized in practice), an appropriate expansion in such a basis (whether re= al or complex) should give the true ground state.
  2. "that usi= ng complex valued orbitals would not reach a lower energy state" -= this is forbidden by the variation theorem,=20 the validity of=20 which does not rely upon the=20 real or complex nature of the basis.
  3. "Are there any kinds o= f chemical bonds that exist for complex-valued=20 molecular orbitals that do not exist for real-valued molecular orbitals?" -=20 A linear combination of=20 degenerate eigenfunctions of=20 a Hermitian operator=20 will also be an eigenfunction with the same eigenvalue. So an appropriate l= inear combination of degenerate complex orbitals can be constructed to give= a real orbital=20 with the same energy. For example, if the Hamiltonian is real, psi and psi*= will be degenerate, and psi+psi* will give a real orbital with the same en= ergy.
  4. The=20 symmetry labels sigma, pi, delta,... are strictly valid only for linear mol= ecules. Since molecular orbitals typically extend over multiple centers, th= e appropriate symmetry designations to use for the molecular orbitals will = be the irreducible representations of the point group of the molecule.
    <= /li>
N. Sukumar
Profess= or of Chemistry
Director, Center for Informatics
Shiv Nadar Un= iversity, India
https://chemistry.snu.edu.in/people/faculty/n-sukumar
<= span>"The ability to reduce everything to simple fundamental laws does= not imply the ability to start from those laws and reconstruct the univers= e."
- Phillip W. Anderson, in "More is Different" (1972)<= /span>
<= /div>


On Tue, Sep 22,= 2020 at 11:54 AM Thomas Manz thomasamanz**gma= il.com <owner-chemistry() c= cl.net> wrote:
Dear colleagues,

In my research, = I've encountered a question about real-valued versus complex-valued mol= ecular orbitals. Normally, most software=C2=A0programs that perform DFT, Ha= rtree-Fock, coupled=C2=A0cluster, or configuration interaction type quantum= chemistry computations on molecules use real-valued molecular=C2=A0orbital= s.=C2=A0

My first question is whether there exists= any theorem that shows this will converge to the ground state (i.e., that = using complex valued orbitals would not reach a lower energy state) when th= e Hamiltonian does not contain any spin-orbit coupling or applied magnetic = field? In other words, when the multi-electronic Hamiltonian is comprised o= f the normal terms: electron kinetic energy, nuclear-electron potential ene= rgy, nuclear-nuclear potential energy, electron-electron Coulomb & exch= ange-correlation energies.

My second question is r= egarding periodic DFT calculations for which many software programs use com= plex-valued molecular orbitals. Are their any kinds of chemical bonds that = exist for complex-valued molecular orbitals that do not exist for real-valu= ed molecular orbitals? Are their any new bonding symmetries made possible f= or the complex-valued orbitals that cannot exist for the real-valued orbita= ls? For real-valued molecular orbitals, the primary covalent bond-orbital s= ymmetries are sigma, pi, delta, and phi. Do complex-valued orbitals enable = any additional bond-orbital symmetries?

I sincerel= y appreciate any insights into this topic you can provide.

Tom
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