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Electron Charge Density: A Clue from Quantum Chemistry for Quantum Foundations

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Abstract

Within quantum chemistry, the electron clouds that surround nuclei in atoms and molecules are sometimes treated as clouds of probability and sometimes as clouds of charge. These two roles, tracing back to Schrödinger and Born, are in tension with one another but are not incompatible. Schrödinger’s idea that the nucleus of an atom is surrounded by a spread-out electron charge density is supported by a variety of evidence from quantum chemistry, including two methods that are used to determine atomic and molecular structure: the Hartree-Fock method and density functional theory. Taking this evidence as a clue to the foundations of quantum physics, Schrödinger’s electron charge density can be incorporated into many different interpretations of quantum mechanics (and extensions of such interpretations to quantum field theory).

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Notes

  1. This is also how Born [15] retells the story. Other authors explicitly challenge the common narrative and defend the potential value of Schrödinger’s original charge density role (e.g., Einstein [16, p. 168–169]; Jaynes [17]; Bader [18,19,20]; Bell [21, p. 39–40]; Bacciagaluppi and Valentini [22, Chap. 4]; Allori et al. [23]; Norsen [24, Chap. 5]; Gao [6, 7]).

  2. See also [25, Sect. 3.6]; [26, Chap. 17]; [27, p. 10, 147, 460].

  3. Similar examples appear in [29, p. 151]; [18, p. 6]; [30, p. 771]; [31, p. 1141]; [32, p. 223]; [27, p. 403, 460].

  4. Szabo and Ostlund [29, Sect. 3.1.1] sound like they may be thinking along the same lines when they contrast the Coulomb potential sourced by an electron’s “instantaneous position” with the Coulomb potential arrived at by taking a psi-squared-weighted average over the electron’s possible positions. Nelson [33] compares the above understanding of amplitude-squared as describing time-averaged particle motion to a number of alternatives.

  5. In quantum chemistry, see (for example) [29, Sect. 3.8.3]; [18, Sect. 1.3]; [20]; [31]; [27, p. 403].

  6. The Hartree-Fock Method can also be used for excited states [29, Sect. 2.2.6], but we will not discuss that application here.

  7. See, for example, [35, Chap. 9]; [36, Sect. 17.2]; [37, p. 432]; [29, Sect. 2.3.6]; [27, Sect. 11.1].

  8. See [29, Sect. 2.1.2]; [28, Sect. 10.1]; [32, Chap. 8]; [27, Sect. 13.1].

  9. For certain combinations of atoms, there will be multiple local minima corresponding to different stable arrangements of the nuclei (structural isomers and stereoisomers). See [38]; [27, Sect. 15.10]; [39].

  10. This Hamiltonian appears in [27, p. 345].

  11. Modifying the Hartree-Fock method, you can use the relativistic Dirac equation in place of the non-relativistic Schrödinger equation. With that modification, it is called the “Dirac-Fock” or “Dirac-Hartree-Fock” method [40]; [27, p. 581].

  12. Usually, it is assumed that each of these spin orbitals can be written as the product of a single component complex-valued spatial wave function and a two-component spinor that has no spatial dependence. In the restricted Hartree-Fock method (which can be used when there are an even number of electrons), one adds the assumption that electrons come in pairs having exactly the same spatial wave function and opposite spins (see [29, Chap. 3]; [41, p. 12]; [32, Sect. 7.17]).

  13. For discussion of how good this approximation is, see [36, Chap. 18]; [42, p. 164]; [27, Sect. 11.1].

  14. The interpretation of correlation energy is discussed in [43, Sect. II.C]; [37, p. 436]; [28, Sect. 9.8].

  15. This method is presented in [29, Chap. 4]; [28, Sect. 9.8]; [32, Sect. 9.6]; [27, Sect. 16.2]

  16. See [31, 44,45,46].

  17. See footnote 28.

  18. In classical electrostatics, the density of potential energy for a distribution of charge \(\rho ^q(\vec {x})\) in a potential \(V(\vec {x})\) can be written as \(\frac{1}{2}\rho ^q(\vec {x}) V(\vec {x})\) [47, Sect. 1.11]; [48, Sect. 2.4]. Because there will be an equal contribution to the potential energy from the nuclei in the potential sourced by the electron charge density \(\rho ^q(\vec {x})\), (16) does not contain the factor of \(\frac{1}{2}\). Moving from electrostatics to electromagnetism, this potential energy of charged matter is replaced by electromagnetic field energy [49, Chap. 5]; [48, Sect. 2.4.4]. However, for our purposes here (where we are only concerned with electrostatic Coulomb attraction and repulsion), it will be convenient to treat this kind of energy as potential energy possessed by matter.

  19. See [18, p. 9]; [27, problem 16.28].

  20. Note that the availability of this interpretation is independent of the Hartree-Fock approximation. For any wave function (Slater determinant or not), the contribution to the expectation value of the energy from the third term in (8) is equal to the classical electrostatic energy of the Schrödinger charge distribution (6) derived from that wave function interacting with the point charge nuclei.

  21. The derivation of (17)–(21) from the electron-electron interaction term in (8) appears in [37].

  22. By treating the nuclei as point charges and only considering contributions to the energy from pairs of distinct nuclei, we have also left out any contributions from nuclear self-repulsion.

  23. Schrödinger [13, Sect. 3] applied his charge density role for amplitude-squared to the interpretation of such Coulomb integrals when analyzing multi-electron atoms. In the included discussion with Born, Heisenberg, and Fowler, Schrödinger briefly discusses Hartree’s method (a precursor to Hartree-Fock that leaves out the exchange integrals) in order to show that his “hope of achieving a three-dimensional conception [a physical understanding of what is happening in three-dimensional space] is not quite utopian,” though he notes the remaining challenge of incorporating and interpreting exchange terms [22, p. 428–429].

  24. This way of rearranging terms is mentioned in [37, p. 436]; [50, p. 5048]; [41, p. 7].

  25. See [27, Sect. 9.4 and 11.1].

  26. McQuarrie [28, p. 482–489] reviews in detail a similar Hartree-Fock calculation of the ground state of the helium atom, summing over just two terms instead of the five in (24).

  27. Calculations of the bond angle of \(\hbox {H}_2\)O have been discussed in the literature on scientific realism as a point of agreement between the different interpretations of quantum mechanics [52, p. S309]; [53, Sect. 4.5.2].

  28. As Levine [27, Sect. 15.8] explains, the Hartree-Fock method does not (and cannot) yield a unique set of orbitals. There are alternative sets of orbitals that lead to the same multi-electron wave function when you take the Slater determinant.

  29. See [57, p. 69]; [58].

  30. Although Levine [27, Sect. 16.5] calls the electron density an “electron probability density,” he does not mean by this that integrating the electron density (7) over any spatial region gives the probability of finding at least one electron in that region. For sufficiently small regions, the probability of finding more than one electron in the region is negligible and the integrated electron density over that region can be interpreted as giving the probability of finding one (or at least one) electron in that region. However, for large regions the possibility of finding multiple electrons cannot be neglected. The integral of the electron density (7) over all space is N (the total number of electrons) not 1 (the probability of finding at least one electron if you look everywhere). Thus, the electron density may be interpreted as an expected number density (as was explained in Sect. 2) but, speaking precisely, it cannot be interpreted as a probability density.

  31. This way of writing the energy functional matches [41, Eq. 3.2.3]; [32, Eq. 9.36]; [27, Eq. 16.36], though here I have also included the potential energy associated with repulsion between the nuclei to parallel the presentation of the Hartree-Fock method in Sect. 3.1 (as in [60, Eq. 6.12]).

  32. This extension is explained clearly in [62, p. 5384].

  33. See [41, Eq. 7.1.1 & 7.1.13]; [62, Eq. 3.14]; [64, Eq. 5.11]; [60, Eq. 7.5]; [32, p. 320]; [58, Eq. 18]; [27, Eq. 16.44].

  34. For more on the self-interaction correction, see [50]; [41, Sect. 8.3]; [62, 65]; [57, footnote 33]; [64, Sect. 2.3 & 6.7]; [61, Sect. 4.7]; [66].

  35. These applications are discussed in [67]; [60, Sect. 8.9]; [68]; [27, p. 571].

  36. See [11, Sect. 2]; [12, p. 1066–1068]; [69, Sect. 3.1]; [29, Sect. 3.4.7]; [70]; [20, p. 2]; [22, Sect. 4.4]; [27, Sect. 14.2].

  37. See also [71]; [72]; [69, p. 40]; [41, Sect. 1.6]; [25, Sect. 6.2]; [19]; [20, Sect. 4.1].

  38. A third option, GRWf, adds to the ontology special points in spacetime where collapse events (flashes) occur. One could potentially associate these events with momentarily existing point particles that have both mass and charge. Or, one could formulate GRWf so that spacetime contains neither mass nor charge.

  39. See also [78, p. 38]; [79]. Ghirardi et al. [76, Sect. 4.3] consider adding a charge density in addition to a mass density, but reject the idea for reasons that are particular to the continuous spontaneous localization theory that they are discussing (and would not apply to GRW).

  40. For discussion of this supposed virtue of GRWm, see [79, 82,83,84].

  41. One might also wonder whether the matter density of GRWm acts a source for the gravitational field. Derakhshani [92] has explored the idea that it does, which would provide support for calling the matter density a “mass density.”

  42. Because there is no electrostatic self-interaction, in Sect. 3.1 the potential energy of each electron would be determined by only looking at the electromagnetic fields sourced by the other particles. This raises awkward questions as to whether there are many electromagnetic fields or just one [49, p. 36, 158].

  43. In addition to densities of mass and charge, GRWmc (and Smc below) should also include densities of momentum and current to describe the flow of mass and charge.

  44. Another option would be GRWc, with only a charge density and no mass density. (A potential problem for GRWc is mentioned by Allori et al. in [23, footnote 1].)

  45. For reasons that we can put aside here, Stern-Gerlach spin measurements for individual charged particles are difficult and neutral atoms are used instead [93, p. 230]; [94, p. 181]; [95, Sect. 2].

  46. One might similarly argue that Schrödinger’s charge density is already present, deflating the disagreement between S0, Sm, and Smc.

  47. In a realistic scenario, the wave function would branch into many pieces that could be sorted into a z-spin up cluster and a z-spin down cluster [107, Sect. 9], [97, 108, Sect. 3.11]. For our purposes here, we can idealize and speak of two branches post-measurement.

  48. Carroll [109, Chap. 8] makes similar moves regarding the status of energy in the multiverse as a whole and energy within each branch.

  49. See [110, p. 153]; [111, p. 154].

  50. See [115,116,117]; [33, p. 645]; [118]; [7, footnote 5].

  51. Gao [6, Chap. 8] combines this “random discontinuous motion of particles” with a collapse dynamics for the wave function. Without this wave function collapse, his proposal would be a many-worlds theory: Bell’s Everett (?) theory [119]; [120, Sect. 5.1]; [75, Sect. 6.2]; [23, Sect. 4]; [121]; [6, Chap. 8].

  52. Similar approaches are discussed in [122,123,124,125,126].

  53. It would be unwise to assign each electron a smaller charge so that the total electron charge (across all worlds) for the helium atom is what you’d expect, \(-2e\). One problem is that the branching that occurs upon measurement within this interpretation is a process where clusters of worlds separate from one another (in configuration space), leading to a dimming like that discussed for the many-worlds interpretation. For example, if you decide to move your helium atom (with total electron charge \(-2e\)) left or right depending on the result of a quantum measurement with two equiprobable outcomes, then once that is done the total electron charge of the atom will be \(-e\) (not \(-2e\)) in the cluster of worlds where the atom was moved to the left and will also be \(-e\) (not \(-2e\)) in the cluster where the atom was moved to the right.

  54. See [127,128,129] and references therein.

  55. Ultimately, it may be best to view classical fermion fields (like the Dirac field) as Grassmann-valued, not complex-valued. The reasons for this and the challenges that accompany such a picture will not be discussed here (see [127, 130,131,132,133,134,135,136]). The problems with Grassmann field values have led some to advocate only using wave functionals for bosons [137, 138].

  56. I have recently studied this classical field description of electrons in a series of papers focused on understanding electron spin as the actual rotation of energy and charge [95, 136, 139, 140].

  57. See [133, Chap. 4]; [127, 134, 135].

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Acknowledgements

Thank you to Craig Callender, Eddy Keming Chen, Scott Cushing, Maaneli Derakhshani, Mario Hubert, Joshua Hunt, Gerald Knizia, Logan McCarty, and Eric Winsberg for helpful feedback and discussion.

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    Charles T. Sebens

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Sebens, C.T. Electron Charge Density: A Clue from Quantum Chemistry for Quantum Foundations. Found Phys 51, 75 (2021). https://doi.org/10.1007/s10701-021-00480-7

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