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I remember when I was taking quantum chemistry, and I learned that every molecule’s physical configuration is realizable as a very simple looking Hamiltonian (t

I remember when I was taking quantum chemistry, and I learned that every molecule’s physical configuration is realizable as a very simple looking Hamiltonian (the typical charge potential + kinetic terms). How is it possible that this simple little equation accounts for all (non-relativistic) chemical properties and geometries?? For example, consider a water molecule: 10 protons, 10 electrons, and 8 neutrons. The Hamiltonian for this molecule consists of the pairwise interactions of the charges along with the kinetic terms for the individual particles, and that’s it. You don’t distinguish between the “atoms” in the molecule. Then, it’s too hard to calculate directly (we typically use various approximations, e.g., Born-Oppenheimer), but if you were able to explicitly compute the energy eigenstates for the full molecular (Coulomb) Hamiltonian, you could take these state vectors to generate probability distributions over physical particle configurations, and you would find that the “clusters” of eigenstates that correspond to energy eigenvalues yield the actual physical shapes of the water molecule. That’s mind-blowing! And this applies to any molecule at all (or physical system). If, for a specific piece of DNA, you were able to compute the energy eigenstates of the Hamiltonian corresponding to the total number of protons, neutrons, and electrons for that DNA strand (just three integers used to generate the form of the entire molecular Hamiltonian), you would find that somewhere very (very) far along in the landscape of energy eigenvalues that your particular piece of DNA and its geometric form exists. My multiple attempts to share with friends and family my excitement about the beauty of such a simple equation encoding the physical manifestation of the entirety of chemistry were met with blank stares and a change of the subject, so yes, the well-being benefit to admiring nature’s beauty probably applies only if you’re in a relationship with another scientist.

dekhn 5y ago
if you actually do quantum chemistry you will see the math is not simple. Your comments about DNA are basically wrong, as well, as the structure of DNA is defined partly by its environment (the specific details of waters that are only weakly bound, or not bound at all, salts and other ions).
- Xcelerate 5y ago
  > if you actually do quantum chemistry I have—for my PhD dissertation I performed QMC calculations for lithium intercalation in carbon. Why do you assume I haven't? > as the structure of DNA is defined partly by its environment (the specific details of waters that are only weakly bound... Yes. Of course I've done water box simulations. Should I elaborate on all the minutiae of molecular dynamics for the reader? Or am I allowed to make some simplifications? I'll be more precise: suppose we consider a strand of DNA and its immediate surrounding environment, and trace out the rest of the universe to produce a density matrix of just that subsystem. This is a solution to the Schrodinger equation for the entire subsystem, and we consider the resultant (not configurational) Hamiltonian for that system without any adiabatic approximations, making sure to specify a basis that removes symmetries (rotational, translational, permutational, etc.). Then you will find that if you could actually perform such a diagonalization of that massive Hamiltonian (I am quite aware that you can't), then a density plot of # eigenvalues vs energy will have peaks, and these peaks will correspond to real spatial structures of DNA, as previously mentioned. If you want to precisely specify "spacial structure", then let's use the definition of Salas, et al. (https://arxiv.org/pdf/2109.04542.pdf https://arxiv.org/pdf/2109.04542.pdf): > A generally unambiguous concept of shape is straightforward if we return to the original spatial probability density in (3N − 3)-dimensional configuration space. Maxima of this function will correspond to relatively likely configurations. We will refer to the arrangement of such maxima in the configuration space as the quantal shape of the system. Furthermore, I don't see how anything I said was particularly controversial, since I'm just paraphrasing others in the field. See R.F. Hendry in "Philosophy of Chemistry: Synthesis of a New Discipline", where he essentially says what I said above more precisely: The resultant Hamiltonian for any particular system is that which arises from the following method: i) Specify a list of fundamental physical interactions (gravitational, electormagnetic, strong- and weak-nuclear). ii) Enumerate the microparticles present in the relevant system and list their charges, masses, and values of any other relevant quantities. iii) Using only the approved “fundamental” forces in (i), list the interactions occuurring between the microparticles enumerated in (ii) iv) Using the results of steps (i)-(iii), write down the kinetic and potential energy operators and add them Happily for the strict physicalist, something very much like this method is taught to every student of quantum chemistry, with the proviso that particle enumeration remains at the level of electrons and nuclei, and only electrostatic terms are usually included in the potential energy operator. It really does yield a Hamiltonian for every molecule.

dekhn 5y ago

if you actually do quantum chemistry you will see the math is not simple. Your comments about DNA are basically wrong, as well, as the structure of DNA is defined partly by its environment (the specific details of waters that are only weakly bound, or not bound at all, salts and other ions).

Xcelerate 5y ago
> if you actually do quantum chemistry I have—for my PhD dissertation I performed QMC calculations for lithium intercalation in carbon. Why do you assume I haven't? > as the structure of DNA is defined partly by its environment (the specific details of waters that are only weakly bound... Yes. Of course I've done water box simulations. Should I elaborate on all the minutiae of molecular dynamics for the reader? Or am I allowed to make some simplifications? I'll be more precise: suppose we consider a strand of DNA and its immediate surrounding environment, and trace out the rest of the universe to produce a density matrix of just that subsystem. This is a solution to the Schrodinger equation for the entire subsystem, and we consider the resultant (not configurational) Hamiltonian for that system without any adiabatic approximations, making sure to specify a basis that removes symmetries (rotational, translational, permutational, etc.). Then you will find that if you could actually perform such a diagonalization of that massive Hamiltonian (I am quite aware that you can't), then a density plot of # eigenvalues vs energy will have peaks, and these peaks will correspond to real spatial structures of DNA, as previously mentioned. If you want to precisely specify "spacial structure", then let's use the definition of Salas, et al. (https://arxiv.org/pdf/2109.04542.pdf https://arxiv.org/pdf/2109.04542.pdf): > A generally unambiguous concept of shape is straightforward if we return to the original spatial probability density in (3N − 3)-dimensional configuration space. Maxima of this function will correspond to relatively likely configurations. We will refer to the arrangement of such maxima in the configuration space as the quantal shape of the system. Furthermore, I don't see how anything I said was particularly controversial, since I'm just paraphrasing others in the field. See R.F. Hendry in "Philosophy of Chemistry: Synthesis of a New Discipline", where he essentially says what I said above more precisely: The resultant Hamiltonian for any particular system is that which arises from the following method: i) Specify a list of fundamental physical interactions (gravitational, electormagnetic, strong- and weak-nuclear). ii) Enumerate the microparticles present in the relevant system and list their charges, masses, and values of any other relevant quantities. iii) Using only the approved “fundamental” forces in (i), list the interactions occuurring between the microparticles enumerated in (ii) iv) Using the results of steps (i)-(iii), write down the kinetic and potential energy operators and add them Happily for the strict physicalist, something very much like this method is taught to every student of quantum chemistry, with the proviso that particle enumeration remains at the level of electrons and nuclei, and only electrostatic terms are usually included in the potential energy operator. It really does yield a Hamiltonian for every molecule.