Dear colleagues,

In my research, I've encountered a question about real-valued versus c= omplex-valued molecular orbitals. Normally, most software programs tha= t perform DFT, Hartree-Fock, coupled cluster, or configuration interac= tion 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 contai= n any spin-orbit coupling or applied magnetic field? In other words, when the multi-electronic Hamiltonian is c= omprised of the normal terms: electron kinetic energy, nuclear-electron pot= ential energy, nuclear-nuclear potential energy, electron-electron Coulomb = & exchange-correlation energies.

My second question is regarding periodic DFT calculations for which ma= ny software programs use complex-valued molecular orbitals. Are their any k= inds of chemical bonds that exist for complex-valued molecular orbitals tha= t do not exist for real-valued molecular orbitals? Are their any new bonding symmetries made possible for the compl= ex-valued orbitals that cannot exist for the real-valued orbitals? For real= -valued molecular orbitals, the primary covalent bond-orbital symmetries ar= e 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

--_000_9c006cbc0da44379b1947679c63e7b79chtumde_-- 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:

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.
"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.
"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.
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

--000000000000a60d0205afe2ee23-- From owner-chemistry@ccl.net Tue Sep 22 12:33:01 2020 From: "quapp=mathematik.uni-leipzig.de" To: CCL Subject: CCL: real-valued versus complex-valued molecular orbitals Message-Id: <-54172-200922061107-15006-/mBlnVKS9GMp3Wrr8f6Vjw]^[server.ccl.net> X-Original-From: quapp-$-mathematik.uni-leipzig.de Content-Transfer-Encoding: 8bit Content-Type: multipart/alternative; boundary="=_3OkF3gygKffrfD9Ww8WP3AL" Date: Tue, 22 Sep 2020 12:10:59 +0200 MIME-Version: 1.0 Sent to CCL by: quapp#%#mathematik.uni-leipzig.de This message is in MIME format. --=_3OkF3gygKffrfD9Ww8WP3AL Content-Type: text/plain; charset=utf-8; format=flowed; DelSp=Yes Content-Description: Textnachricht Content-Disposition: inline Content-Transfer-Encoding: 8bit Dear Tom, if it helps? Note that the Schrödinger Eq. starts with the imaginary i. Thus all 'Quantum mechanics' gives complex-valued results... It is the physical theory which goes intrinsically thrugh the complex numbers. Regards Wolfgang ----- Nachricht von "Thomas Manz thomasamanz**gmail.com" --------- Datum: Mon, 21 Sep 2020 23:18:08 -0600 Von: "Thomas Manz thomasamanz**gmail.com" Antwort an: CCL Subscribers Betreff: CCL: real-valued versus complex-valued molecular orbitals An: "Quapp, Wolfgang " > 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 ----- Ende der Nachricht von "Thomas Manz thomasamanz**gmail.com" ----- --=_3OkF3gygKffrfD9Ww8WP3AL Content-Type: text/html; charset=utf-8 Content-Description: HTML-Nachricht Content-Disposition: inline
Dear Tom,

if it helps? Note that the Schrödinger Eq. starts with the imaginary i.
Thus all 'Quantum mechanics' gives complex-valued results...
It is the physical theory which goes intrinsically thrugh the complex numbers.

Regards
Wolfgang

----- Nachricht von "Thomas Manz thomasamanz**gmail.com" <owner-chemistry.:.ccl.net> ---------
Datum: Mon, 21 Sep 2020 23:18:08 -0600
Von: "Thomas Manz thomasamanz**gmail.com" <owner-chemistry.:.ccl.net>
Antwort an: CCL Subscribers <chemistry.:.ccl.net>
Betreff: CCL: real-valued versus complex-valued molecular orbitals
An: "Quapp, Wolfgang " <quapp.:.rz.uni-leipzig.de>

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

----- Ende der Nachricht von "Thomas Manz thomasamanz**gmail.com" <owner-chemistry.:.ccl.net> -----

--=_3OkF3gygKffrfD9Ww8WP3AL-- From owner-chemistry@ccl.net Tue Sep 22 14:03:00 2020 From: "Susi Lehtola susi.lehtola-x-helsinki.fi" To: CCL Subject: CCL: real-valued versus complex-valued molecular orbitals Message-Id: <-54173-200922140034-14980-mDV4I6/1+5HZmMjumcl/rQ###server.ccl.net> X-Original-From: Susi Lehtola Content-Transfer-Encoding: 7bit Content-Type: text/plain; charset=US-ASCII Date: Tue, 22 Sep 2020 21:00:22 +0300 MIME-Version: 1.0 Sent to CCL by: Susi Lehtola [susi.lehtola**helsinki.fi] On Mon, 21 Sep 2020 23:18:08 -0600 Thomas Manz thomasamanz**gmail.com 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. Dear Thomas, the usual rationale for real-valued orbitals is that the molecular Hamiltonian does not have any imaginary units or complex conjugates. Then, it is easy to see that if \psi is a solution, then \psi^* is also a solution, and you can take a linear combination of these which is real. However, once you go to quantum chemical models, this is no longer the case: single-determinant theories often break symmetries, which is also the case here. For instance, restricted Hartree-Fock is well-known to exhibit complex instabilities, see e.g. the discussion by Small, Sundstrom and Head-Gordon in doi:10.1063/1.4905120. A hand-waving argument is that if you do a natural orbital analysis of the real part of the resulting density matrix, you'll see that it has eigenvalues that differ from 0 and 2, that is, it would correspond to a multi-determinental real-valued wave function. You can also find cases where spin-restricted Kohn-Sham theory has complex instabilities, see Lee et al in doi:10.1103/PhysRevLett.123.113001. > 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. The key in periodic DFT calculations is the Bloch theorem, which states that the orbitals have a periodicity which is inherently complex-valued: \psi_k (r + g) = u_k(r) exp(i k . g) where k is the crystal momentum and g is a lattice vector. Because of this, I'm not sure calculations with real-valued crystalline orbitals would even make sense. I don't think that there would be a difference between kinds of chemical bonds between real-valued and complex-valued calculations. The bond-orbital symmetry classification originates from diatomic molecules, where the exact orbitals have the symmetry (see e.g. my recent review doi:10.1002/qua.25968) \psi (r) = \chi(\mu,\nu) exp(i m \varphi) where \varphi is the cylindrical angle measured around the bond, and \mu and \nu are coordinates of the prolate spheroidal coordinate system. m is an integer, which defines the orbital symmetry: m=0 for sigma, |m|=1 for pi, |m|=2 for delta, and |m|=3 for phi orbitals. The m dependence is only in the \varphi part. Real-valued vs complex-valued orbitals allow you to mix the different m components; e.g. instead of exp(i m \varphi) you could get cos(m \varphi) and sin(m \varphi), but these still have the same symmetry: both cos (m \varphi) and sin (m \varphi) only have contributions from m=m. This means that if your model breaks the complex conjugation symmetry i.e. you have a complex instability, an analysis in terms of sigma, pi, delta, phi etc orbitals should still be possible and make sense. A way more important aspect is again the symmetry breaking: e.g. the HF and DFT ground states of the CH radical have sigma-delta symmetry breaking, where the occupied orbitals don't have the proper symmetry any more, as there will be mixing between m values. Hope this helps. Susi -- ------------------------------------------------------------------ Mr. Susi Lehtola, PhD Junior Fellow, Adjunct Professor susi.lehtola(-)helsinki.fi University of Helsinki http://susilehtola.github.io/ Finland ------------------------------------------------------------------ Susi Lehtola, dosentti, FT tutkijatohtori susi.lehtola(-)helsinki.fi Helsingin yliopisto http://susilehtola.github.io/ ------------------------------------------------------------------ From owner-chemistry@ccl.net Tue Sep 22 15:13:01 2020 From: "Vitaly Rassolov rassolov[#]mailbox.sc.edu" To: CCL Subject: CCL: real-valued versus complex-valued molecular orbitals Message-Id: <-54174-200922151126-11732-r3fEcZkCLh6ZmpRIIL7BKw###server.ccl.net> X-Original-From: "Vitaly Rassolov" Date: Tue, 22 Sep 2020 15:11:24 -0400 Sent to CCL by: "Vitaly Rassolov" [rassolov=mailbox.sc.edu] Dear Tom, Regarding the complex-valued orbitals: this depends on the level of electron correlation. Two statements are easy to prove: The ground state of a non-relativistic Hamiltonian can always chosen to be real, provided there are no constraints to the wavefunction form. I strongly suspect that this remains true in a finite one-particle basis set limit. Once we limit the wavefunction to be a single determinant, there are (many) cases when complex orbitals yield lower energy (search for the Generalized Hartree-Fock for the examples). Again, I strongly suspect this is true for any finite CI or coupled cluster expansion. Regarding the symmetry question: I don't know the answer and hope that someone in the CCL can shed a light on it. My only, not directly relevant, remark is that the complex "p+" or "p-" orbitals are not degenerate to the "p0" or the real-valued "px", "py" orbitals even in atoms, once we include electron-electron interactions: the spatial distributions of electrons on these orbitals are different. However, I suspect that possible different symmetries in real and complex bonds are visible only in the two-particle space (which is not the kind of space most computational chemists contemplate). Vitaly Rassolov Department of Chemistry and Biochemistry University of South Carolina 631 Sumter St, Columbia SC 29208 https://sc.edu/study/colleges_schools/chemistry_and_biochemistry/our_people/directory/rassolov_vitaly.php > "Thomas Manz thomasamanz**gmail.com" wrote: > > Sent to CCL by: Thomas Manz [thomasamanz]|[gmail.com] > --0000000000005661b105afe018ce > Content-Type: text/plain; charset="UTF-8" > > 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 > > --0000000000005661b105afe018ce > Content-Type: text/html; charset="UTF-8" > Content-Transfer-Encoding: quoted-printable > >
Dear colleagues,

In my research, I'= > ve encountered a question about real-valued versus complex-valued molecular= > orbitals. Normally, most software=C2=A0programs that perform DFT, Hartree-= > Fock, coupled=C2=A0cluster, or configuration interaction type quantum chemi= > stry computations on molecules use real-valued molecular=C2=A0orbitals.=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 the Ham= > iltonian 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 regard= > ing 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 mo= > lecular orbitals? Are their any new bonding symmetries made possible for th= > e complex-valued orbitals that cannot exist for the real-valued orbitals? F= > or real-valued molecular orbitals, the primary covalent bond-orbital symmet= > ries are sigma, pi, delta, and phi. Do complex-valued orbitals enable any a= > dditional bond-orbital symmetries?

I sincerely app= > reciate any insights into this topic you can provide.

<= > div>Tom

> > --0000000000005661b105afe018ce-- > >