8054:). It is easy to show that for two bodies in circular orbits around their center of mass (regardless of individual masses), the momenta of the two bodies are equal and opposite, and that for any collection of particles in the center-of-mass frame, the net momentum is zero. Given that the center-of-mass frame is the lab frame (where the molecule is stationary), the momentum of the nuclei must be equal and opposite to that of the electrons. A hand-waving justification can be derived from quantum mechanics as well. The corresponding operators do not contain mass and the molecule can be treated as a
4024:
7954:. A nice fit between the symmetry-preserved single-state treatment and the corresponding two-state treatment was obtained. This applies in particular to the reactive state-to-state probabilities (see Table III in Ref. 5a and Table III in Ref. 5b), for which the ordinary BO approximation led to erroneous results, whereas the symmetry-preserving BO approximation produced the accurate results, as they followed from solving the two coupled equations.
4741:
1471:
3162:
3700:
4448:
4437:
1188:
1960:
2560:
4019:{\displaystyle T_{\text{n}}(\mathbf {R} )_{k'k}\equiv {\big (}\mathbb {H} _{\text{n}}(\mathbf {R} ){\big )}_{k'k}=\delta _{k'k}T_{\text{n}}-\sum _{A,\alpha }{\frac {1}{M_{A}}}\langle \chi _{k'}|P_{A\alpha }|\chi _{k}\rangle _{(\mathbf {r} )}P_{A\alpha }+\langle \chi _{k'}|T_{\text{n}}|\chi _{k}\rangle _{(\mathbf {r} )}.}
2949:
81:
The approximation is widely used in quantum chemistry to speed up the computation of molecular wavefunctions and other properties for large molecules. There are cases where the assumption of separable motion no longer holds, which make the approximation lose validity (it is said to "break down"), but
2183:
242:
problem then consists only of the 126 electronic coordinates. This electronic computation is then repeated for other possible positions of the nuclei, i.e. deformations of the molecule. For benzene, this could be done using a grid of 36 possible nuclear position coordinates. The electronic energies
4736:{\displaystyle \langle \chi _{k'}||\chi _{k}\rangle _{(\mathbf {r} )}=iZ_{A}\sum _{i}\left\langle \chi _{k'}\left|{\frac {(\mathbf {r} _{iA})_{\alpha }}{r_{iA}^{3}}}\right|\chi _{k}\right\rangle _{(\mathbf {r} )}\quad {\text{with}}\quad \mathbf {r} _{iA}\equiv \mathbf {r} _{i}-\mathbf {R} _{A}.}
69:
in a molecule can be treated separately, based on the fact that the nuclei are much heavier than the electrons. Due to the larger relative mass of a nucleus compared to an electron, the coordinates of the nuclei in a system are approximated as fixed, while the coordinates of the electrons are
2906:
6657:
4816:, the nuclear momentum coupling term becomes large and is no longer negligible. This is the case where the BO approximation breaks down, and a coupled set of nuclear motion equations must be considered instead of the one equation appearing in the second step of the BO approximation.
7938:
8122:
736:
6118:
5925:
2770:
173:
1466:{\displaystyle H_{\text{e}}=-\sum _{i}{{\frac {1}{2}}\nabla _{i}^{2}}-\sum _{i,A}{\frac {Z_{A}}{r_{iA}}}+\sum _{i>j}{\frac {1}{r_{ij}}}+\sum _{B>A}{\frac {Z_{A}Z_{B}}{R_{AB}}}\quad {\text{and}}\quad T_{\text{n}}=-\sum _{A}{{\frac {1}{2M_{A}}}\nabla _{A}^{2}}.}
1120:
7080:
4204:
1793:
2410:
418:
mass and the masses of atomic nuclei, and correspondingly the time scales of their motion. Given the same amount of momentum, the nuclei move much more slowly than the electrons. In mathematical terms, the BO approximation consists of expressing the
7683:
3653:
Solution of these coupled equations gives an approximation for energy and wavefunction that goes beyond the Born–Oppenheimer approximation. Unfortunately, the off-diagonal kinetic energy terms are usually difficult to handle. This is why often a
5032:
2351:
is a molecular orbital (MO) given as a linear expansion of atomic orbitals (AOs). An AO depends visibly on the coordinates of an electron, but the nuclear coordinates are not explicit in the MO. However, upon change of geometry, i.e., change of
6237:
583:
8330:
Baer, Roi; Charutz, David M.; Kosloff, Ronnie; Baer, Michael (22 November 1996). "A study of conical intersection effects on scattering processes: The validity of adiabatic single-surface approximations within a quasi-Jahn–Teller model".
1014:. To that end the second step of the BO approximation is generalized to a set of coupled eigenvalue equations depending on nuclear coordinates only. Off-diagonal elements in these equations are shown to be nuclear kinetic energy terms.
3157:{\displaystyle {\big (}\mathbb {H} _{\text{e}}(\mathbf {R} ){\big )}_{k'k}\equiv \langle \chi _{k'}(\mathbf {r} ;\mathbf {R} )|H_{\text{e}}|\chi _{k}(\mathbf {r} ;\mathbf {R} )\rangle _{(\mathbf {r} )}=\delta _{k'k}E_{k}(\mathbf {R} )}
1761:
We assume that the molecule is in a homogeneous (no external force) and isotropic (no external torque) space. The only interactions are the two-body
Coulomb interactions among the electrons and nuclei. The Hamiltonian is expressed in
5372:
3434:
8091:, that the same electronic state (for instance, the electronic ground state) is obtained upon small changes of the nuclear geometry. The method would give a discontinuity (jump) in the PES if electronic state switching would occur.
4894:
squared and, accordingly, is then negligible also. Only the first (diagonal) kinetic energy term in this equation survives in the case of well separated surfaces, and a diagonal, uncoupled, set of nuclear motion equations results:
3658:
transformation is applied, which retains part of the nuclear kinetic energy terms on the diagonal, removes the kinetic energy terms from the off-diagonal and creates coupling terms between the adiabatic PESs on the off-diagonal.
2048:
6446:
5677:
3311:
7532:
7342:
4139:
5040:
We reiterate that when two or more potential energy surfaces approach each other, or even cross, the Born–Oppenheimer approximation breaks down, and one must fall back on the coupled equations. Usually one invokes then the
3648:
1614:
3662:
If we can neglect the off-diagonal elements the equations will uncouple and simplify drastically. In order to show when this neglect is justified, we suppress the coordinates in the notation and write, by applying the
6296:
7946:
The relevance of the present approach was demonstrated while studying a two-arrangement-channel model (containing one inelastic channel and one reactive channel) for which the two adiabatic states were coupled by a
202:
consist of kinetic energies, interelectronic repulsions, internuclear repulsions, and electron–nuclear attractions, which are the terms typically included when computing the electronic structure of molecules.
4814:
96:
8100:
This equation is time-independent, and stationary wavefunctions for the nuclei are obtained; nevertheless, it is traditional to use the word "motion" in this context, although classically motion implies time
7733:
2781:
1688:
6909:
6914:
6841:
638:
3216:
985:
5931:
5738:
3535:
1517:
3575:
2651:
2013:
3470:
2659:
1766:, so that we do not see Planck's constant, the dielectric constant of the vacuum, electronic charge, or electronic mass in this formula. The only constants explicitly entering the formula are
1023:
7943:
In this equation the inhomogeneity ensures the symmetry for the perturbed part of the solution along any contour and therefore for the solution in the required region in configuration space.
2605:
778:
7581:
4432:{\displaystyle \langle \chi _{k'}|P_{A\alpha }|\chi _{k}\rangle _{(\mathbf {r} )}={\frac {\langle \chi _{k'}||\chi _{k}\rangle _{(\mathbf {r} )}}{E_{k}(\mathbf {R} )-E_{k'}(\mathbf {R} )}}.}
1010:
It will be discussed how the BO approximation may be derived and under which conditions it is applicable. At the same time we will show how the BO approximation may be improved by including
5602:
8374:
Adhikari, Satrajit; Billing, Gert D. (1999). "The conical intersection effects and adiabatic single-surface approximations on scattering processes: A time-dependent wave packet approach".
6741:
7725:
6430:
1177:
8417:
Charutz, David M.; Baer, Roi; Baer, Michael (1997). "A study of degenerate vibronic coupling effects on scattering processes: are resonances affected by degenerate vibronic coupling?".
5477:
1955:{\displaystyle T_{\text{n}}=\sum _{A}\sum _{\alpha =x,y,z}{\frac {P_{A\alpha }P_{A\alpha }}{2M_{A}}}\quad {\text{with}}\quad P_{A\alpha }=-i{\frac {\partial }{\partial R_{A\alpha }}}.}
2555:{\displaystyle P_{A\alpha }\chi _{k}(\mathbf {r} ;\mathbf {R} )=-i{\frac {\partial \chi _{k}(\mathbf {r} ;\mathbf {R} )}{\partial R_{A\alpha }}}\quad {\text{for}}\quad \alpha =x,y,z,}
462:
5444:
8066:), it follows that, indeed, the kinetic energy of the nuclei in a molecule is usually much smaller than the kinetic energy of the electrons, the mass ratio being on the order of 10.
7426:
200:
8471:
7159:
7121:
6779:
6697:
6372:
6334:
5715:
5515:
5406:
5267:
5229:
5191:
5149:
5111:
865:
329:
589:
into electronic and nuclear terms, where cross-terms between electrons and nuclei are neglected, so that the two smaller and decoupled systems can be solved more efficiently.
7589:
4892:
4849:
285:
228:
2937:
867:. Because this procedure of recomputing the electronic wave functions as a function of an infinitesimally changing nuclear geometry is reminiscent of the conditions for the
4901:
5569:
5073:
4169:
3692:
3238:
2372:
2315:
2293:
2040:
998:
is the total energy of the molecule, including contributions from electrons, nuclear vibrations, and overall rotation and translation of the molecule. In accord with the
990:
is solved. This second step of the BO approximation involves separation of vibrational, translational, and rotational motions. This can be achieved by application of the
366:
7189:
4196:
4058:
2399:
2349:
2271:
2244:
2213:
7385:
7365:
6126:
5543:
475:
1756:
1726:
1017:
It will be shown that the BO approximation can be trusted whenever the PESs, obtained from the solution of the electronic Schrödinger equation, are well separated:
393:
614:
the nuclear positions are no longer variable, but are constant parameters (they enter the equation "parametrically"). The electron–nucleus interactions are
227:
in the three-dimensional coordinates of the nuclei and electrons, giving 3 × 12 = 36 nuclear + 3 × 42 = 126 electronic = 162 variables for the wave function. The
5278:
3326:
5053:
To include the correct symmetry within the Born–Oppenheimer (BO) approximation, a molecular system presented in terms of (mass-dependent) nuclear coordinates
2178:{\displaystyle H_{\text{e}}\chi _{k}(\mathbf {r} ;\mathbf {R} )=E_{k}(\mathbf {R} )\chi _{k}(\mathbf {r} ;\mathbf {R} )\quad {\text{for}}\quad k=1,\ldots ,K.}
1002:, the nuclear potential is taken to be an average over electron configurations of the sum of the electron–nuclear and internuclear electric potentials.
7085:
In order for this equation to yield a solution with the correct symmetry, it is suggested to apply a perturbation approach based on an elastic potential
5610:
3246:
7434:
7197:
4063:
8287:
Baer, Michael; Englman, Robert (1997). "A modified Born-Oppenheimer equation: application to conical intersections and other types of singularities".
406:, using it to express the mean force on the nuclei caused by the electrons and thereby skipping the calculation of the nuclear Schrödinger equation.
1522:
5732:
To study the scattering process taking place on the two lowest surfaces, one extracts from the above BO equation the two corresponding equations:
3664:
3584:
6246:
238:
Schrödinger equation is solved, while treating the nuclei as stationary (not "coupled" with the dynamics of the electrons). This corresponding
8271:
7968:
6652:{\displaystyle -{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\chi +({\tilde {u}}_{1}-E)\chi +i{\frac {\hbar ^{2}}{2m}}\chi +i(u_{1}-u_{2})\psi _{2}=0.}
8239:
6911:. Consequently, the last term is, indeed, negligibly small at every point in the region of interest, and the equation simplifies to become
5193:
becomes a closed PES in the region of interest, with the exception of sporadic infinitesimal sites surrounding degeneracy points formed by
175:
These terms are of different orders of magnitude and the nuclear spin energy is so small that it is often omitted. The electronic energies
7933:{\displaystyle -{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\eta +({\tilde {u}}_{1}-E)\eta +i{\frac {\hbar ^{2}}{2m}}\eta =(u_{1}-u_{0})\chi _{0}.}
2901:{\displaystyle \langle \chi _{k'}(\mathbf {r} ;\mathbf {R} )|\chi _{k}(\mathbf {r} ;\mathbf {R} )\rangle _{(\mathbf {r} )}=\delta _{k'k},}
231:, i.e., the computational power required to solve an eigenvalue equation, increases faster than the square of the number of coordinates.
8201:
1623:
4752:
731:{\displaystyle H_{\text{e}}(\mathbf {r} ,\mathbf {R} )\chi (\mathbf {r} ,\mathbf {R} )=E_{\text{e}}\chi (\mathbf {r} ,\mathbf {R} )}
622:
of the nuclei clamped at certain positions in space. (This first step of the BO approximation is therefore often referred to as the
6846:
6784:
6113:{\displaystyle -{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\psi _{2}+({\tilde {u}}_{2}-E)\psi _{2}+{\frac {\hbar ^{2}}{2m}}\psi _{1}=0,}
5920:{\displaystyle -{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\psi _{1}+({\tilde {u}}_{1}-E)\psi _{1}-{\frac {\hbar ^{2}}{2m}}\psi _{2}=0,}
247:
for the nuclei. This potential is then used for a second Schrödinger equation containing only the 36 coordinates of the nuclei.
586:
899:
8514:
3475:
1479:
2765:{\displaystyle \Psi (\mathbf {R} ,\mathbf {r} )=\sum _{k=1}^{K}\chi _{k}(\mathbf {r} ;\mathbf {R} )\phi _{k}(\mathbf {R} ),}
168:{\displaystyle E_{\text{total}}=E_{\text{electronic}}+E_{\text{vibrational}}+E_{\text{rotational}}+E_{\text{nuclear spin}}.}
3540:
3170:
5726:
2610:
1972:
1115:{\displaystyle E_{0}(\mathbf {R} )\ll E_{1}(\mathbf {R} )\ll E_{2}(\mathbf {R} )\ll \cdots {\text{ for all }}\mathbf {R} }
224:
8075:
Typically, the electronic Schrödinger equation for molecules cannot be solved exactly. Approximation methods include the
7978:
3442:
999:
234:
When applying the BO approximation, two smaller, consecutive steps can be used: For a given position of the nuclei, the
7983:
4819:
Conversely, if all surfaces are well separated, all off-diagonal terms can be neglected, and hence the whole matrix of
2571:
744:
8476:
7540:
7075:{\displaystyle -{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\chi +({\tilde {u}}_{1}-E)\chi +i{\frac {\hbar ^{2}}{2m}}\chi =0.}
2404:
We will assume that the parametric dependence is continuous and differentiable, so that it is meaningful to consider
78:, the latter of whom proposed it in 1927 during a period of intense fervent in the development of quantum mechanics.
5574:
1787:
It is useful to introduce the total nuclear momentum and to rewrite the nuclear kinetic energy operator as follows:
8499:
7367:
is an arbitrary contour, and the exponential function contains the relevant symmetry as created while moving along
6702:
2374:, the LCAO coefficients obtain different values and we see corresponding changes in the functional form of the MO
7691:
6383:
1136:
5449:
57:
is the best-known mathematical approximation in molecular dynamics. Specifically, it is the assumption that the
6781:
becomes classically allowed (which happens at the vicinity of the (1, 2) degeneracy points) this implies that:
826:
244:
426:
7164:
The equation with an elastic potential can be solved, in a straightforward manner, by substitution. Thus, if
5411:
3650:
are further discussed below. The vibronic coupling in this approach is through nuclear kinetic energy terms.
8504:
7393:
5042:
3655:
811:
630:
469:
396:
216:
7948:
250:
So, taking the most optimistic estimate for the complexity, instead of a large equation requiring at least
178:
5718:
7126:
7088:
6746:
6664:
6339:
6301:
5685:
5482:
5380:
5234:
5196:
5158:
5116:
5078:
2246:
on the nuclear coordinates is indicated by the symbol after the semicolon. This indicates that, although
832:
290:
7678:{\displaystyle \chi (\mathbf {q} |\Gamma )=\chi _{0}(\mathbf {q} |\Gamma )+\eta (\mathbf {q} |\Gamma ),}
604:
75:
2940:
5027:{\displaystyle \phi _{k}(\mathbf {R} )=E\phi _{k}(\mathbf {R} )\quad {\text{for}}\quad k=1,\ldots ,K,}
8426:
8383:
8340:
8296:
8164:
7963:
7951:
4865:
4822:
253:
2914:
8076:
465:
8007:
Authors often justify this step by stating that "the heavy nuclei move more slowly than the light
5552:
5056:
4851:
is effectively zero. The third term on the right side of the expression for the matrix element of
4144:
3670:
3221:
2355:
2298:
2276:
2018:
2215:
will be taken to be real, which is possible when there are no magnetic or spin interactions. The
403:
338:
31:
6661:
The last term in this equation can be deleted for the following reasons: At those points where
6232:{\displaystyle {\tilde {u}}_{k}(\mathbf {q} )=u_{k}(\mathbf {q} )+(\hbar ^{2}/2m)\tau _{12}^{2}}
578:{\displaystyle \Psi _{\mathrm {total} }=\psi _{\mathrm {electronic} }\psi _{\mathrm {nuclear} }}
7167:
4174:
2377:
2327:
2249:
2222:
2191:
8509:
8442:
8399:
8356:
8312:
8267:
8197:
8088:
8055:
2321:
1011:
991:
868:
619:
42:
38:
17:
8481:
7370:
7350:
5528:
8434:
8391:
8348:
8304:
8259:
8172:
7973:
1731:
1701:
371:
5367:{\displaystyle -{\frac {\hbar ^{2}}{2m}}(\nabla +\tau )^{2}\Psi +(\mathbf {u} -E)\Psi =0,}
4032:
335:
being the number of grid points for the potential) and a very small calculation requiring
82:
even then the approximation is usually used as a starting point for more refined methods.
8256:
Beyond Born-Oppenheimer: Electronic
Nonadiabatic Coupling Terms and Conical Intersections
3429:{\displaystyle {\boldsymbol {\phi }}(\mathbf {R} )=E{\boldsymbol {\phi }}(\mathbf {R} ).}
8430:
8387:
8344:
8300:
8168:
4746:
The matrix element of the one-electron operator appearing on the right side is finite.
1617:
593:
62:
8438:
8308:
5479:
is a diagonal matrix containing the corresponding adiabatic potential energy surfaces
2943:, is over electronic coordinates only. By definition, the matrix with general element
93:
approximation means considering molecular energy as a sum of independent terms, e.g.:
8493:
5722:
5272:
The starting point is the nuclear adiabatic BO (matrix) equation written in the form
58:
7988:
5672:{\displaystyle \mathbf {\tau } _{jk}=\langle \zeta _{j}|\nabla \zeta _{k}\rangle .}
3306:{\displaystyle H\Psi (\mathbf {R} ,\mathbf {r} )=E\Psi (\mathbf {R} ,\mathbf {r} )}
1763:
420:
220:
86:
7527:{\displaystyle -{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\xi _{0}+(u_{0}-E)\xi _{0}=0.}
7337:{\displaystyle \chi _{0}(\mathbf {q} |\Gamma )=\xi _{0}(\mathbf {q} )\exp \left,}
4134:{\displaystyle \langle \chi _{k}|P_{A\alpha }|\chi _{k}\rangle _{(\mathbf {r} )}}
3577:
is diagonal, and the nuclear
Hamilton matrix is non-diagonal; its off-diagonal (
8465:
8258:. Hoboken, NJ, USA: John Wiley & Sons, Inc. 28 March 2006. pp. 26–57.
368:
steps can be performed. In practice, the scaling of the problem is larger than
1609:{\displaystyle \mathbf {R} \equiv \{\mathbf {R} _{A}=(R_{Ax},R_{Ay},R_{Az})\}}
793:
239:
8446:
8403:
8360:
8316:
8177:
8152:
8234:
2320:
For example, in the molecular-orbital-linear-combination-of-atomic-orbitals
464:) of a molecule as the product of an electronic wavefunction and a nuclear (
8263:
3643:{\displaystyle {\big (}\mathbb {H} _{\text{n}}(\mathbf {R} ){\big )}_{k'k}}
287:
hypothetical calculation steps, a series of smaller calculations requiring
5604:
is a matrix containing the vectorial non-adiabatic coupling terms (NACT):
5151:
is considered. To ensure the validity of the BO approximation, the energy
5075:
and formed by the two lowest BO adiabatic potential energy surfaces (PES)
4198:
can be chosen to be always real. The off-diagonal matrix elements satisfy
27:
The notion that the motion of atomic nuclei and electrons can be separated
8230:
8012:
8008:
6291:{\displaystyle \mathbf {\tau } _{12}=\mathbf {\tau } _{12}(\mathbf {q} )}
5546:
415:
71:
66:
8018:
of electrons and nuclei is of the same order of magnitude. In that case
6443:
and combining it with the first equation yields the (complex) equation
893:) is reintroduced, and the Schrödinger equation for the nuclear motion
882:
In the second step of the BO approximation, the nuclear kinetic energy
212:
5037:
which are the normal second step of the BO equations discussed above.
8395:
8352:
8194:
American
Prometheus: The Triumph and Tragedy of J. Robert Oppenheimer
7428:
can be shown to be a solution of the (unperturbed/elastic) equation
780:
is the electronic wavefunction for given positions of nuclei (fixed
8254:"Born-Oppenheimer Approach: Diabatization and Topological Matrix".
7583:, the full solution of the above decoupled equation takes the form
3320:
coupled eigenvalue equations depending on nuclear coordinates only
889:(containing partial derivatives with respect to the components of
402:
The slope of the potential energy surface can be used to simulate
5408:
is a column vector containing the unknown nuclear wave functions
414:
The BO approximation recognizes the large difference between the
1683:{\displaystyle r_{iA}\equiv |\mathbf {r} _{i}-\mathbf {R} _{A}|}
8461:
Resources related to the Born–Oppenheimer approximation:
4809:{\displaystyle E_{k}(\mathbf {R} )\approx E_{k'}(\mathbf {R} )}
3218:
from the left and integration over the electronic coordinates
6904:{\displaystyle u_{1}(\mathbf {q} )-u_{2}(\mathbf {q} )\sim 0}
6298:
is the (vectorial) NACT responsible for the coupling between
871:, this manner of obtaining a PES is often referred to as the
6836:{\displaystyle u_{1}(\mathbf {q} )\sim u_{2}(\mathbf {q} )}
1130:
non-relativistic, time-independent molecular
Hamiltonian:
399:
to further reduce the number of variables and dimensions.
8477:
Another version of the same translation by S. M. Blinder
5049:
The Born–Oppenheimer approximation with correct symmetry
980:{\displaystyle \phi (\mathbf {R} )=E\phi (\mathbf {R} )}
8220:, 3rd ed., MIT Press, Cambridge, MA, 2009, § 28.2.
8216:
T. H. Cormen, C. E. Leiserson, R. L. Rivest, C. Stein,
8011:". Classically, this statement makes sense only if the
3530:{\displaystyle \phi _{k}(\mathbf {R} ),\ k=1,\ldots ,K}
3167:
is diagonal. After multiplication by the real function
1512:{\displaystyle \mathbf {r} \equiv \{\mathbf {r} _{i}\}}
4171:
vanish, because we assume time-reversal invariant, so
7736:
7694:
7592:
7543:
7437:
7396:
7373:
7353:
7200:
7191:
is the solution of this equation, it is presented as
7170:
7129:
7091:
6917:
6849:
6787:
6749:
6705:
6667:
6449:
6386:
6342:
6304:
6249:
6129:
5934:
5741:
5688:
5613:
5577:
5555:
5531:
5485:
5452:
5414:
5383:
5281:
5237:
5199:
5161:
5119:
5081:
5059:
4904:
4868:
4825:
4755:
4451:
4207:
4177:
4147:
4066:
4035:
3703:
3673:
3587:
3570:{\displaystyle \mathbb {H} _{\text{e}}(\mathbf {R} )}
3543:
3478:
3445:
3329:
3249:
3224:
3211:{\displaystyle \chi _{k'}(\mathbf {r} ;\mathbf {R} )}
3173:
2952:
2917:
2784:
2662:
2613:
2574:
2413:
2380:
2358:
2330:
2301:
2279:
2252:
2225:
2194:
2051:
2021:
1975:
1796:
1734:
1704:
1626:
1525:
1482:
1191:
1139:
1026:
902:
835:
810:
in small steps and repeatedly solving the electronic
747:
641:
478:
429:
374:
341:
293:
256:
215:
molecule consists of 12 nuclei and 42 electrons. The
181:
99:
2646:{\displaystyle \chi _{k}(\mathbf {r} ;\mathbf {R} )}
2008:{\displaystyle \chi _{k}(\mathbf {r} ;\mathbf {R} )}
3465:{\displaystyle {\boldsymbol {\phi }}(\mathbf {R} )}
792:for all nuclear coordinates. The electronic energy
7932:
7719:
7677:
7575:
7526:
7420:
7379:
7359:
7336:
7183:
7153:
7115:
7074:
6903:
6835:
6773:
6735:
6691:
6651:
6424:
6366:
6328:
6290:
6231:
6112:
5919:
5709:
5671:
5596:
5563:
5537:
5509:
5471:
5438:
5400:
5366:
5261:
5223:
5185:
5155:of the system is assumed to be low enough so that
5143:
5105:
5067:
5026:
4886:
4843:
4808:
4735:
4431:
4190:
4163:
4133:
4052:
4018:
3686:
3642:
3569:
3529:
3464:
3428:
3305:
3232:
3210:
3156:
2931:
2900:
2764:
2645:
2599:
2554:
2393:
2366:
2343:
2309:
2287:
2265:
2238:
2207:
2177:
2034:
2007:
1954:
1750:
1720:
1682:
1608:
1511:
1465:
1171:
1114:
979:
859:
772:
730:
596:is neglected, that is, the corresponding operator
577:
456:
387:
360:
323:
279:
194:
167:
5549:operator with respect to the nuclear coordinates
7727:satisfies the resulting inhomogeneous equation:
4862:) can approximately be written as the matrix of
2600:{\displaystyle \Psi (\mathbf {R} ,\mathbf {r} )}
773:{\displaystyle \chi (\mathbf {r} ,\mathbf {R} )}
8196:(1st ed.). Vintage Books. pp. 65–66.
7576:{\displaystyle \chi _{0}(\mathbf {q} |\Gamma )}
6435:and the corresponding rearrangements are made:
2939:indicates that the integration, implied by the
8484:, a section from Peter Haynes' doctoral thesis
8155:[On the Quantum Theory of Molecules].
5597:{\displaystyle \mathbf {\tau } (\mathbf {q} )}
1616:of the nuclei are with respect to a Cartesian
618:removed, i.e., the electrons still "feel" the
6736:{\displaystyle \psi _{2}(\mathbf {q} )\sim 0}
3776:
3745:
3621:
3590:
2986:
2955:
1620:. Distances between particles are written as
8:
5704:
5663:
5632:
4525:
4452:
4354:
4283:
4260:
4208:
4114:
4067:
3996:
3947:
3914:
3862:
3667:for differentiation, the matrix elements of
3095:
3008:
2857:
2785:
1603:
1534:
1506:
1491:
8146:
8144:
7720:{\displaystyle \eta (\mathbf {q} |\Gamma )}
6425:{\displaystyle \chi =\psi _{1}+i\psi _{2},}
1172:{\displaystyle H=H_{\text{e}}+T_{\text{n}}}
5472:{\displaystyle \mathbf {u} (\mathbf {q} )}
5269:(designated as (1, 2) degeneracy points).
1519:of the electrons and the position vectors
788:stands for all electronic coordinates and
607:. In the remaining electronic Hamiltonian
243:on this grid are then connected to give a
8176:
7921:
7908:
7895:
7873:
7868:
7852:
7847:
7824:
7818:
7794:
7783:
7782:
7766:
7746:
7740:
7735:
7706:
7701:
7693:
7661:
7656:
7636:
7631:
7622:
7604:
7599:
7591:
7562:
7557:
7548:
7542:
7512:
7493:
7477:
7467:
7447:
7441:
7436:
7410:
7401:
7395:
7372:
7352:
7315:
7306:
7297:
7285:
7275:
7246:
7237:
7219:
7214:
7205:
7199:
7175:
7169:
7143:
7134:
7128:
7105:
7096:
7090:
7054:
7049:
7033:
7028:
7005:
6999:
6975:
6964:
6963:
6947:
6927:
6921:
6916:
6887:
6878:
6863:
6854:
6848:
6825:
6816:
6801:
6792:
6786:
6763:
6754:
6748:
6743:by definition, and at those points where
6719:
6710:
6704:
6681:
6672:
6666:
6637:
6624:
6611:
6586:
6581:
6565:
6560:
6537:
6531:
6507:
6496:
6495:
6479:
6459:
6453:
6448:
6413:
6397:
6385:
6356:
6347:
6341:
6318:
6309:
6303:
6280:
6271:
6266:
6256:
6251:
6248:
6223:
6218:
6200:
6194:
6176:
6167:
6152:
6143:
6132:
6131:
6128:
6095:
6082:
6077:
6061:
6056:
6033:
6027:
6018:
5999:
5988:
5987:
5974:
5964:
5944:
5938:
5933:
5902:
5889:
5884:
5868:
5863:
5840:
5834:
5825:
5806:
5795:
5794:
5781:
5771:
5751:
5745:
5740:
5698:
5689:
5687:
5657:
5645:
5639:
5620:
5615:
5612:
5586:
5578:
5576:
5556:
5554:
5530:
5499:
5490:
5484:
5461:
5453:
5451:
5428:
5419:
5413:
5390:
5382:
5338:
5323:
5291:
5285:
5280:
5251:
5242:
5236:
5213:
5204:
5198:
5175:
5166:
5160:
5133:
5124:
5118:
5095:
5086:
5080:
5060:
5058:
4994:
4985:
4976:
4958:
4949:
4934:
4925:
4912:
4903:
4878:
4873:
4867:
4835:
4830:
4824:
4798:
4784:
4769:
4760:
4754:
4724:
4719:
4709:
4704:
4691:
4686:
4679:
4668:
4664:
4653:
4637:
4629:
4618:
4605:
4600:
4593:
4578:
4562:
4552:
4532:
4528:
4518:
4509:
4499:
4498:
4482:
4470:
4459:
4450:
4415:
4401:
4386:
4377:
4361:
4357:
4347:
4338:
4329:
4313:
4301:
4290:
4280:
4267:
4263:
4253:
4244:
4235:
4226:
4215:
4206:
4182:
4176:
4152:
4146:
4121:
4117:
4107:
4098:
4089:
4080:
4074:
4065:
4034:
4003:
3999:
3989:
3980:
3974:
3965:
3954:
3935:
3921:
3917:
3907:
3898:
3889:
3880:
3869:
3854:
3845:
3833:
3820:
3802:
3781:
3775:
3774:
3765:
3756:
3752:
3751:
3744:
3743:
3726:
3717:
3708:
3702:
3678:
3672:
3626:
3620:
3619:
3610:
3601:
3597:
3596:
3589:
3588:
3586:
3559:
3550:
3546:
3545:
3542:
3492:
3483:
3477:
3454:
3446:
3444:
3415:
3407:
3393:
3385:
3374:
3365:
3361:
3360:
3348:
3339:
3335:
3334:
3328:
3295:
3287:
3267:
3259:
3248:
3225:
3223:
3200:
3192:
3178:
3172:
3146:
3137:
3119:
3102:
3098:
3086:
3078:
3069:
3060:
3054:
3045:
3037:
3029:
3015:
2991:
2985:
2984:
2975:
2966:
2962:
2961:
2954:
2953:
2951:
2921:
2916:
2881:
2864:
2860:
2848:
2840:
2831:
2822:
2814:
2806:
2792:
2783:
2751:
2742:
2730:
2722:
2713:
2703:
2692:
2677:
2669:
2661:
2635:
2627:
2618:
2612:
2589:
2581:
2573:
2522:
2509:
2492:
2484:
2475:
2465:
2448:
2440:
2431:
2418:
2412:
2385:
2379:
2359:
2357:
2335:
2329:
2302:
2300:
2280:
2278:
2257:
2251:
2230:
2224:
2199:
2193:
2145:
2136:
2128:
2119:
2107:
2098:
2083:
2075:
2066:
2056:
2050:
2026:
2020:
1997:
1989:
1980:
1974:
1937:
1924:
1906:
1896:
1886:
1868:
1855:
1848:
1824:
1814:
1801:
1795:
1739:
1733:
1709:
1703:
1675:
1669:
1664:
1654:
1649:
1643:
1631:
1625:
1591:
1575:
1559:
1543:
1538:
1526:
1524:
1500:
1495:
1483:
1481:
1453:
1448:
1435:
1422:
1421:
1415:
1399:
1389:
1377:
1366:
1356:
1349:
1337:
1319:
1310:
1298:
1280:
1270:
1264:
1252:
1238:
1233:
1219:
1218:
1212:
1196:
1190:
1163:
1150:
1138:
1107:
1102:
1088:
1079:
1064:
1055:
1040:
1031:
1025:
969:
949:
932:
923:
910:
901:
849:
840:
834:
784:), is solved approximately. The quantity
762:
754:
746:
720:
712:
700:
685:
677:
663:
655:
646:
640:
550:
549:
511:
510:
484:
483:
477:
435:
434:
428:
395:, and more approximations are applied in
379:
373:
346:
340:
317:
313:
298:
292:
273:
261:
255:
186:
180:
156:
143:
130:
117:
104:
98:
8482:The Born–Oppenheimer approximation
8151:Max Born; J. Robert Oppenheimer (1927).
1780:– the atomic number and mass of nucleus
457:{\displaystyle \Psi _{\mathrm {total} }}
225:partial differential eigenvalue equation
223:and wavefunction of this molecule, is a
8113:
8056:box containing the electrons and nuclei
8000:
7821:
7743:
7444:
7002:
6924:
6534:
6456:
6191:
6030:
5941:
5837:
5748:
5439:{\displaystyle \psi _{k}(\mathbf {q} )}
5288:
4442:The matrix element in the numerator is
3447:
3408:
3386:
806:of the nuclei. Varying these positions
8192:Bird, Kai; Sherwin, Martin K. (2006).
8087:It is assumed, in accordance with the
7421:{\displaystyle \xi _{0}(\mathbf {q} )}
7969:Adiabatic process (quantum mechanics)
219:, which must be solved to obtain the
195:{\displaystyle E_{\text{electronic}}}
74:and his 23-year-old graduate student
70:dynamic. The approach is named after
7:
8243:. New York: Oxford University Press.
8240:Dynamical Theory of Crystal Lattices
8129:. Chemical Education Digital Library
8123:"The Born-Oppenheimer Approximation"
4860:Born–Oppenheimer diagonal correction
7154:{\displaystyle u_{1}(\mathbf {q} )}
7116:{\displaystyle u_{0}(\mathbf {q} )}
6774:{\displaystyle u_{2}(\mathbf {q} )}
6692:{\displaystyle u_{2}(\mathbf {q} )}
6439:Multiplying the second equation by
6377:Next a new function is introduced:
6367:{\displaystyle u_{2}(\mathbf {q} )}
6329:{\displaystyle u_{1}(\mathbf {q} )}
5710:{\displaystyle |\zeta _{n}\rangle }
5525:is the total energy of the system,
5521:is the reduced mass of the nuclei,
5510:{\displaystyle u_{k}(\mathbf {q} )}
5401:{\displaystyle \Psi (\mathbf {q} )}
5262:{\displaystyle u_{2}(\mathbf {q} )}
5224:{\displaystyle u_{1}(\mathbf {q} )}
5186:{\displaystyle u_{2}(\mathbf {q} )}
5144:{\displaystyle u_{2}(\mathbf {q} )}
5106:{\displaystyle u_{1}(\mathbf {q} )}
2565:which in general will not be zero.
1698:) and similar definitions hold for
860:{\displaystyle E_{e}(\mathbf {R} )}
585:. This enables a separation of the
324:{\displaystyle 126^{2}N=15\,876\,N}
7864:
7858:
7763:
7711:
7666:
7641:
7609:
7567:
7464:
7374:
7354:
7320:
7276:
7224:
7045:
7039:
6944:
6577:
6571:
6476:
6073:
6067:
5961:
5880:
5874:
5768:
5650:
5532:
5384:
5352:
5329:
5310:
4749:When the two surfaces come close,
4500:
3281:
3253:
2663:
2575:
2502:
2468:
1930:
1926:
1445:
1230:
569:
566:
563:
560:
557:
554:
551:
539:
536:
533:
530:
527:
524:
521:
518:
515:
512:
497:
494:
491:
488:
485:
480:
448:
445:
442:
439:
436:
431:
25:
8339:(20). AIP Publishing: 9141–9152.
8153:"Zur Quantentheorie der Molekeln"
2295:, its functional form depends on
7702:
7657:
7632:
7600:
7558:
7411:
7307:
7286:
7247:
7215:
7144:
7106:
6888:
6864:
6826:
6802:
6764:
6720:
6682:
6357:
6319:
6281:
6177:
6153:
5587:
5557:
5500:
5462:
5454:
5429:
5391:
5339:
5252:
5214:
5176:
5134:
5096:
5061:
4986:
4959:
4935:
4799:
4770:
4720:
4705:
4687:
4669:
4601:
4533:
4416:
4387:
4362:
4268:
4122:
4004:
3922:
3766:
3718:
3611:
3560:
3493:
3455:
3416:
3394:
3375:
3349:
3296:
3288:
3268:
3260:
3226:
3201:
3193:
3147:
3103:
3087:
3079:
3038:
3030:
2976:
2922:
2865:
2849:
2841:
2815:
2807:
2752:
2731:
2723:
2678:
2670:
2636:
2628:
2590:
2582:
2493:
2485:
2449:
2441:
2360:
2303:
2281:
2137:
2129:
2108:
2084:
2076:
1998:
1990:
1665:
1650:
1539:
1527:
1496:
1484:
1108:
1089:
1065:
1041:
970:
950:
933:
875:and the PES itself is called an
850:
802:depends on the chosen positions
763:
755:
721:
713:
686:
678:
664:
656:
8376:The Journal of Chemical Physics
8333:The Journal of Chemical Physics
4999:
4993:
4887:{\displaystyle P_{\alpha }^{A}}
4844:{\displaystyle P_{\alpha }^{A}}
4684:
4678:
3240:the total Schrödinger equation
2527:
2521:
2150:
2144:
1901:
1895:
1394:
1388:
592:In the first step, the nuclear
280:{\displaystyle 162^{2}=26\,244}
8058:. Since the kinetic energy is
7914:
7888:
7879:
7840:
7806:
7788:
7778:
7714:
7707:
7698:
7669:
7662:
7653:
7644:
7637:
7628:
7612:
7605:
7596:
7570:
7563:
7554:
7505:
7486:
7415:
7407:
7323:
7316:
7302:
7251:
7243:
7227:
7220:
7211:
7148:
7140:
7110:
7102:
7060:
7021:
6987:
6969:
6959:
6892:
6884:
6868:
6860:
6830:
6822:
6806:
6798:
6768:
6760:
6724:
6716:
6686:
6678:
6630:
6604:
6592:
6553:
6519:
6501:
6491:
6361:
6353:
6323:
6315:
6285:
6277:
6211:
6187:
6181:
6173:
6157:
6149:
6137:
6088:
6049:
6011:
5993:
5983:
5895:
5856:
5818:
5800:
5790:
5690:
5646:
5591:
5583:
5504:
5496:
5466:
5458:
5433:
5425:
5395:
5387:
5349:
5335:
5320:
5307:
5256:
5248:
5218:
5210:
5180:
5172:
5138:
5130:
5100:
5092:
4990:
4982:
4963:
4955:
4942:
4939:
4931:
4905:
4803:
4795:
4774:
4766:
4673:
4665:
4615:
4596:
4537:
4529:
4510:
4506:
4475:
4471:
4420:
4412:
4391:
4383:
4366:
4358:
4339:
4335:
4306:
4302:
4272:
4264:
4245:
4227:
4126:
4118:
4099:
4081:
4008:
4000:
3981:
3966:
3926:
3918:
3899:
3881:
3770:
3762:
3723:
3714:
3615:
3607:
3564:
3556:
3497:
3489:
3459:
3451:
3420:
3412:
3398:
3390:
3382:
3379:
3371:
3353:
3345:
3330:
3300:
3284:
3272:
3256:
3205:
3189:
3151:
3143:
3107:
3099:
3091:
3075:
3061:
3046:
3042:
3026:
2980:
2972:
2932:{\displaystyle (\mathbf {r} )}
2926:
2918:
2869:
2861:
2853:
2837:
2823:
2819:
2803:
2756:
2748:
2735:
2719:
2682:
2666:
2640:
2624:
2594:
2578:
2497:
2481:
2453:
2437:
2188:The electronic wave functions
2141:
2125:
2112:
2104:
2088:
2072:
2002:
1986:
1676:
1644:
1600:
1552:
1093:
1085:
1069:
1061:
1045:
1037:
974:
966:
954:
946:
940:
937:
929:
903:
854:
846:
767:
751:
725:
709:
690:
674:
668:
652:
18:Born-Oppenheimer approximation
1:
8439:10.1016/s0009-2614(96)01494-7
8309:10.1016/s0009-2614(96)01411-x
8295:(1–2). Elsevier BV: 105–108.
2273:is a real-valued function of
603:is subtracted from the total
8472:Translation by S. M. Blinder
8382:(1). AIP Publishing: 40–47.
5564:{\displaystyle \mathbf {q} }
5068:{\displaystyle \mathbf {q} }
4164:{\displaystyle P_{A\alpha }}
3687:{\displaystyle T_{\text{n}}}
3233:{\displaystyle \mathbf {r} }
2367:{\displaystyle \mathbf {R} }
2310:{\displaystyle \mathbf {R} }
2288:{\displaystyle \mathbf {r} }
2035:{\displaystyle H_{\text{e}}}
30:Not to be confused with the
8425:(6). Elsevier BV: 629–637.
5721:assumed to form a complete
1690:(distance between electron
361:{\displaystyle 36^{2}=1296}
8531:
8218:Introduction to Algorithms
7161:at the asymptotic region.
5717:are eigenfunctions of the
2042:, that is, we have solved
1969:electronic eigenfunctions
29:
7184:{\displaystyle \chi _{0}}
4191:{\displaystyle \chi _{k}}
2394:{\displaystyle \chi _{k}}
2344:{\displaystyle \chi _{k}}
2266:{\displaystyle \chi _{k}}
2239:{\displaystyle \chi _{k}}
2208:{\displaystyle \chi _{k}}
8419:Chemical Physics Letters
8289:Chemical Physics Letters
8178:10.1002/andp.19273892002
7979:Born–Huang approximation
3316:is turned into a set of
2911:and where the subscript
2607:is expanded in terms of
2568:The total wave function
1000:Hellmann–Feynman theorem
827:potential energy surface
245:potential energy surface
229:computational complexity
7984:Franck–Condon principle
7380:{\displaystyle \Gamma }
7360:{\displaystyle \Gamma }
7123:, which coincides with
6699:is classically closed,
5725:in the given region in
5538:{\displaystyle \nabla }
873:adiabatic approximation
397:computational chemistry
8264:10.1002/0471780081.ch2
7934:
7721:
7679:
7577:
7528:
7422:
7381:
7361:
7338:
7185:
7155:
7117:
7076:
6905:
6837:
6775:
6737:
6693:
6653:
6426:
6368:
6330:
6292:
6233:
6114:
5921:
5719:electronic Hamiltonian
5711:
5673:
5598:
5565:
5539:
5511:
5473:
5440:
5402:
5368:
5263:
5225:
5187:
5145:
5107:
5069:
5028:
4888:
4845:
4810:
4737:
4433:
4192:
4165:
4135:
4054:
4020:
3688:
3644:
3571:
3531:
3466:
3430:
3307:
3234:
3212:
3158:
2933:
2902:
2766:
2708:
2647:
2601:
2556:
2395:
2368:
2345:
2311:
2289:
2267:
2240:
2209:
2179:
2036:
2009:
1956:
1752:
1751:{\displaystyle R_{AB}}
1722:
1721:{\displaystyle r_{ij}}
1684:
1610:
1513:
1467:
1173:
1116:
981:
861:
774:
732:
579:
458:
389:
362:
325:
281:
196:
169:
47:Born–Oppenheimer
8515:J. Robert Oppenheimer
7935:
7722:
7680:
7578:
7529:
7423:
7382:
7362:
7339:
7186:
7156:
7118:
7077:
6906:
6838:
6776:
6738:
6694:
6654:
6427:
6369:
6331:
6293:
6234:
6115:
5922:
5712:
5674:
5599:
5566:
5540:
5512:
5474:
5441:
5403:
5369:
5264:
5226:
5188:
5146:
5108:
5070:
5029:
4889:
4846:
4811:
4738:
4434:
4193:
4166:
4136:
4055:
4021:
3689:
3645:
3572:
3532:
3467:
3431:
3308:
3235:
3213:
3159:
2934:
2903:
2767:
2688:
2648:
2602:
2557:
2396:
2369:
2346:
2312:
2290:
2268:
2241:
2217:parametric dependence
2210:
2180:
2037:
2010:
1957:
1753:
1723:
1685:
1611:
1514:
1476:The position vectors
1468:
1174:
1117:
982:
862:
775:
733:
605:molecular Hamiltonian
580:
459:
390:
388:{\displaystyle n^{2}}
363:
326:
282:
197:
170:
76:J. Robert Oppenheimer
8466:The original article
8127:Chemistry Libretexts
7964:Adiabatic ionization
7952:conical intersection
7734:
7692:
7590:
7541:
7435:
7394:
7371:
7351:
7198:
7168:
7127:
7089:
6915:
6847:
6785:
6747:
6703:
6665:
6447:
6384:
6340:
6302:
6247:
6127:
5932:
5739:
5686:
5611:
5575:
5553:
5529:
5483:
5450:
5412:
5381:
5279:
5235:
5197:
5159:
5117:
5079:
5057:
4902:
4866:
4823:
4753:
4449:
4205:
4175:
4145:
4064:
4053:{\displaystyle k'=k}
4033:
3701:
3671:
3585:
3541:
3476:
3443:
3327:
3247:
3222:
3171:
2950:
2915:
2782:
2660:
2611:
2572:
2411:
2378:
2356:
2328:
2299:
2277:
2250:
2223:
2192:
2049:
2019:
1973:
1794:
1732:
1702:
1624:
1523:
1480:
1189:
1137:
1024:
900:
833:
812:Schrödinger equation
745:
639:
631:Schrödinger equation
587:Hamiltonian operator
476:
427:
410:Detailed description
372:
339:
291:
254:
217:Schrödinger equation
179:
97:
8431:1997CPL...265..629C
8388:1999JChPh.111...40A
8345:1996JChPh.105.9141B
8301:1997CPL...265..105B
8169:1927AnP...389..457B
8077:Hartree-Fock method
6228:
5727:configuration space
4883:
4840:
4642:
1458:
1243:
1104: for all
8157:Annalen der Physik
7930:
7717:
7675:
7573:
7524:
7418:
7377:
7357:
7334:
7181:
7151:
7113:
7072:
6901:
6833:
6771:
6733:
6689:
6649:
6422:
6364:
6326:
6288:
6229:
6214:
6110:
5917:
5707:
5669:
5594:
5561:
5535:
5507:
5469:
5436:
5398:
5364:
5259:
5221:
5183:
5141:
5103:
5065:
5024:
4884:
4869:
4841:
4826:
4806:
4733:
4625:
4567:
4429:
4188:
4161:
4131:
4060:) matrix elements
4050:
4016:
3844:
3684:
3640:
3567:
3527:
3462:
3439:The column vector
3426:
3303:
3230:
3208:
3154:
2929:
2898:
2762:
2643:
2597:
2552:
2391:
2364:
2341:
2307:
2285:
2263:
2236:
2205:
2175:
2032:
2005:
1952:
1847:
1819:
1748:
1718:
1680:
1606:
1509:
1463:
1444:
1420:
1348:
1309:
1263:
1229:
1217:
1169:
1126:We start from the
1112:
977:
857:
770:
728:
575:
454:
404:molecular dynamics
385:
358:
321:
277:
192:
165:
32:Born approximation
8500:Quantum chemistry
8273:978-0-471-78008-3
8089:adiabatic theorem
7838:
7791:
7760:
7461:
7019:
6972:
6941:
6551:
6504:
6473:
6140:
6047:
5996:
5958:
5854:
5803:
5765:
5305:
4997:
4915:
4682:
4643:
4558:
4424:
4332:
3977:
3860:
3829:
3823:
3759:
3711:
3681:
3604:
3579:vibronic coupling
3553:
3505:
3368:
3342:
3057:
2969:
2525:
2519:
2219:of the functions
2148:
2059:
2029:
1947:
1899:
1893:
1820:
1810:
1804:
1442:
1411:
1402:
1392:
1386:
1333:
1328:
1294:
1289:
1248:
1227:
1208:
1199:
1166:
1153:
1105:
1012:vibronic coupling
994:. The eigenvalue
992:Eckart conditions
926:
913:
877:adiabatic surface
869:adiabatic theorem
821:as a function of
703:
649:
620:Coulomb potential
189:
159:
146:
133:
120:
107:
43:molecular physics
39:quantum chemistry
16:(Redirected from
8522:
8451:
8450:
8414:
8408:
8407:
8396:10.1063/1.479360
8371:
8365:
8364:
8353:10.1063/1.472748
8327:
8321:
8320:
8284:
8278:
8277:
8251:
8245:
8244:
8227:
8221:
8214:
8208:
8207:
8189:
8183:
8182:
8180:
8148:
8139:
8138:
8136:
8134:
8118:
8102:
8098:
8092:
8085:
8079:
8073:
8067:
8005:
7974:Avoided crossing
7939:
7937:
7936:
7931:
7926:
7925:
7913:
7912:
7900:
7899:
7878:
7877:
7872:
7857:
7856:
7851:
7839:
7837:
7829:
7828:
7819:
7799:
7798:
7793:
7792:
7784:
7771:
7770:
7761:
7759:
7751:
7750:
7741:
7726:
7724:
7723:
7718:
7710:
7705:
7684:
7682:
7681:
7676:
7665:
7660:
7640:
7635:
7627:
7626:
7608:
7603:
7582:
7580:
7579:
7574:
7566:
7561:
7553:
7552:
7533:
7531:
7530:
7525:
7517:
7516:
7498:
7497:
7482:
7481:
7472:
7471:
7462:
7460:
7452:
7451:
7442:
7427:
7425:
7424:
7419:
7414:
7406:
7405:
7386:
7384:
7383:
7378:
7366:
7364:
7363:
7358:
7343:
7341:
7340:
7335:
7330:
7326:
7319:
7314:
7310:
7301:
7293:
7289:
7280:
7279:
7250:
7242:
7241:
7223:
7218:
7210:
7209:
7190:
7188:
7187:
7182:
7180:
7179:
7160:
7158:
7157:
7152:
7147:
7139:
7138:
7122:
7120:
7119:
7114:
7109:
7101:
7100:
7081:
7079:
7078:
7073:
7059:
7058:
7053:
7038:
7037:
7032:
7020:
7018:
7010:
7009:
7000:
6980:
6979:
6974:
6973:
6965:
6952:
6951:
6942:
6940:
6932:
6931:
6922:
6910:
6908:
6907:
6902:
6891:
6883:
6882:
6867:
6859:
6858:
6842:
6840:
6839:
6834:
6829:
6821:
6820:
6805:
6797:
6796:
6780:
6778:
6777:
6772:
6767:
6759:
6758:
6742:
6740:
6739:
6734:
6723:
6715:
6714:
6698:
6696:
6695:
6690:
6685:
6677:
6676:
6658:
6656:
6655:
6650:
6642:
6641:
6629:
6628:
6616:
6615:
6591:
6590:
6585:
6570:
6569:
6564:
6552:
6550:
6542:
6541:
6532:
6512:
6511:
6506:
6505:
6497:
6484:
6483:
6474:
6472:
6464:
6463:
6454:
6431:
6429:
6428:
6423:
6418:
6417:
6402:
6401:
6373:
6371:
6370:
6365:
6360:
6352:
6351:
6335:
6333:
6332:
6327:
6322:
6314:
6313:
6297:
6295:
6294:
6289:
6284:
6276:
6275:
6270:
6261:
6260:
6255:
6238:
6236:
6235:
6230:
6227:
6222:
6204:
6199:
6198:
6180:
6172:
6171:
6156:
6148:
6147:
6142:
6141:
6133:
6119:
6117:
6116:
6111:
6100:
6099:
6087:
6086:
6081:
6066:
6065:
6060:
6048:
6046:
6038:
6037:
6028:
6023:
6022:
6004:
6003:
5998:
5997:
5989:
5979:
5978:
5969:
5968:
5959:
5957:
5949:
5948:
5939:
5926:
5924:
5923:
5918:
5907:
5906:
5894:
5893:
5888:
5873:
5872:
5867:
5855:
5853:
5845:
5844:
5835:
5830:
5829:
5811:
5810:
5805:
5804:
5796:
5786:
5785:
5776:
5775:
5766:
5764:
5756:
5755:
5746:
5716:
5714:
5713:
5708:
5703:
5702:
5693:
5678:
5676:
5675:
5670:
5662:
5661:
5649:
5644:
5643:
5628:
5627:
5619:
5603:
5601:
5600:
5595:
5590:
5582:
5570:
5568:
5567:
5562:
5560:
5544:
5542:
5541:
5536:
5516:
5514:
5513:
5508:
5503:
5495:
5494:
5478:
5476:
5475:
5470:
5465:
5457:
5445:
5443:
5442:
5437:
5432:
5424:
5423:
5407:
5405:
5404:
5399:
5394:
5373:
5371:
5370:
5365:
5342:
5328:
5327:
5306:
5304:
5296:
5295:
5286:
5268:
5266:
5265:
5260:
5255:
5247:
5246:
5230:
5228:
5227:
5222:
5217:
5209:
5208:
5192:
5190:
5189:
5184:
5179:
5171:
5170:
5150:
5148:
5147:
5142:
5137:
5129:
5128:
5112:
5110:
5109:
5104:
5099:
5091:
5090:
5074:
5072:
5071:
5066:
5064:
5033:
5031:
5030:
5025:
4998:
4995:
4989:
4981:
4980:
4962:
4954:
4953:
4938:
4930:
4929:
4917:
4916:
4913:
4893:
4891:
4890:
4885:
4882:
4877:
4850:
4848:
4847:
4842:
4839:
4834:
4815:
4813:
4812:
4807:
4802:
4794:
4793:
4792:
4773:
4765:
4764:
4742:
4740:
4739:
4734:
4729:
4728:
4723:
4714:
4713:
4708:
4699:
4698:
4690:
4683:
4680:
4677:
4676:
4672:
4663:
4659:
4658:
4657:
4648:
4644:
4641:
4636:
4624:
4623:
4622:
4613:
4612:
4604:
4594:
4588:
4587:
4586:
4566:
4557:
4556:
4541:
4540:
4536:
4523:
4522:
4513:
4505:
4504:
4503:
4490:
4489:
4474:
4469:
4468:
4467:
4438:
4436:
4435:
4430:
4425:
4423:
4419:
4411:
4410:
4409:
4390:
4382:
4381:
4371:
4370:
4369:
4365:
4352:
4351:
4342:
4334:
4333:
4330:
4321:
4320:
4305:
4300:
4299:
4298:
4281:
4276:
4275:
4271:
4258:
4257:
4248:
4243:
4242:
4230:
4225:
4224:
4223:
4197:
4195:
4194:
4189:
4187:
4186:
4170:
4168:
4167:
4162:
4160:
4159:
4141:of the operator
4140:
4138:
4137:
4132:
4130:
4129:
4125:
4112:
4111:
4102:
4097:
4096:
4084:
4079:
4078:
4059:
4057:
4056:
4051:
4043:
4025:
4023:
4022:
4017:
4012:
4011:
4007:
3994:
3993:
3984:
3979:
3978:
3975:
3969:
3964:
3963:
3962:
3943:
3942:
3930:
3929:
3925:
3912:
3911:
3902:
3897:
3896:
3884:
3879:
3878:
3877:
3861:
3859:
3858:
3846:
3843:
3825:
3824:
3821:
3815:
3814:
3810:
3794:
3793:
3789:
3780:
3779:
3769:
3761:
3760:
3757:
3755:
3749:
3748:
3739:
3738:
3734:
3721:
3713:
3712:
3709:
3693:
3691:
3690:
3685:
3683:
3682:
3679:
3649:
3647:
3646:
3641:
3639:
3638:
3634:
3625:
3624:
3614:
3606:
3605:
3602:
3600:
3594:
3593:
3576:
3574:
3573:
3568:
3563:
3555:
3554:
3551:
3549:
3536:
3534:
3533:
3528:
3503:
3496:
3488:
3487:
3471:
3469:
3468:
3463:
3458:
3450:
3435:
3433:
3432:
3427:
3419:
3411:
3397:
3389:
3378:
3370:
3369:
3366:
3364:
3352:
3344:
3343:
3340:
3338:
3312:
3310:
3309:
3304:
3299:
3291:
3271:
3263:
3239:
3237:
3236:
3231:
3229:
3217:
3215:
3214:
3209:
3204:
3196:
3188:
3187:
3186:
3163:
3161:
3160:
3155:
3150:
3142:
3141:
3132:
3131:
3127:
3111:
3110:
3106:
3090:
3082:
3074:
3073:
3064:
3059:
3058:
3055:
3049:
3041:
3033:
3025:
3024:
3023:
3004:
3003:
2999:
2990:
2989:
2979:
2971:
2970:
2967:
2965:
2959:
2958:
2941:bra–ket notation
2938:
2936:
2935:
2930:
2925:
2907:
2905:
2904:
2899:
2894:
2893:
2889:
2873:
2872:
2868:
2852:
2844:
2836:
2835:
2826:
2818:
2810:
2802:
2801:
2800:
2771:
2769:
2768:
2763:
2755:
2747:
2746:
2734:
2726:
2718:
2717:
2707:
2702:
2681:
2673:
2652:
2650:
2649:
2644:
2639:
2631:
2623:
2622:
2606:
2604:
2603:
2598:
2593:
2585:
2561:
2559:
2558:
2553:
2526:
2523:
2520:
2518:
2517:
2516:
2500:
2496:
2488:
2480:
2479:
2466:
2452:
2444:
2436:
2435:
2426:
2425:
2400:
2398:
2397:
2392:
2390:
2389:
2373:
2371:
2370:
2365:
2363:
2350:
2348:
2347:
2342:
2340:
2339:
2316:
2314:
2313:
2308:
2306:
2294:
2292:
2291:
2286:
2284:
2272:
2270:
2269:
2264:
2262:
2261:
2245:
2243:
2242:
2237:
2235:
2234:
2214:
2212:
2211:
2206:
2204:
2203:
2184:
2182:
2181:
2176:
2149:
2146:
2140:
2132:
2124:
2123:
2111:
2103:
2102:
2087:
2079:
2071:
2070:
2061:
2060:
2057:
2041:
2039:
2038:
2033:
2031:
2030:
2027:
2014:
2012:
2011:
2006:
2001:
1993:
1985:
1984:
1965:Suppose we have
1961:
1959:
1958:
1953:
1948:
1946:
1945:
1944:
1925:
1914:
1913:
1900:
1897:
1894:
1892:
1891:
1890:
1877:
1876:
1875:
1863:
1862:
1849:
1846:
1818:
1806:
1805:
1802:
1757:
1755:
1754:
1749:
1747:
1746:
1727:
1725:
1724:
1719:
1717:
1716:
1689:
1687:
1686:
1681:
1679:
1674:
1673:
1668:
1659:
1658:
1653:
1647:
1639:
1638:
1615:
1613:
1612:
1607:
1599:
1598:
1583:
1582:
1567:
1566:
1548:
1547:
1542:
1530:
1518:
1516:
1515:
1510:
1505:
1504:
1499:
1487:
1472:
1470:
1469:
1464:
1459:
1457:
1452:
1443:
1441:
1440:
1439:
1423:
1419:
1404:
1403:
1400:
1393:
1390:
1387:
1385:
1384:
1372:
1371:
1370:
1361:
1360:
1350:
1347:
1329:
1327:
1326:
1311:
1308:
1290:
1288:
1287:
1275:
1274:
1265:
1262:
1244:
1242:
1237:
1228:
1220:
1216:
1201:
1200:
1197:
1178:
1176:
1175:
1170:
1168:
1167:
1164:
1155:
1154:
1151:
1121:
1119:
1118:
1113:
1111:
1106:
1103:
1092:
1084:
1083:
1068:
1060:
1059:
1044:
1036:
1035:
986:
984:
983:
978:
973:
953:
936:
928:
927:
924:
915:
914:
911:
866:
864:
863:
858:
853:
845:
844:
779:
777:
776:
771:
766:
758:
737:
735:
734:
729:
724:
716:
705:
704:
701:
689:
681:
667:
659:
651:
650:
647:
626:approximation.)
584:
582:
581:
576:
574:
573:
572:
544:
543:
542:
502:
501:
500:
472:) wavefunction.
463:
461:
460:
455:
453:
452:
451:
394:
392:
391:
386:
384:
383:
367:
365:
364:
359:
351:
350:
330:
328:
327:
322:
303:
302:
286:
284:
283:
278:
266:
265:
201:
199:
198:
193:
191:
190:
187:
174:
172:
171:
166:
161:
160:
157:
148:
147:
144:
135:
134:
131:
122:
121:
118:
109:
108:
105:
21:
8530:
8529:
8525:
8524:
8523:
8521:
8520:
8519:
8490:
8489:
8459:
8454:
8416:
8415:
8411:
8373:
8372:
8368:
8329:
8328:
8324:
8286:
8285:
8281:
8274:
8253:
8252:
8248:
8229:
8228:
8224:
8215:
8211:
8204:
8191:
8190:
8186:
8163:(20): 457–484.
8150:
8149:
8142:
8132:
8130:
8121:Hanson, David.
8120:
8119:
8115:
8111:
8106:
8105:
8099:
8095:
8086:
8082:
8074:
8070:
8053:
8042:
8031:
8024:
8006:
8002:
7997:
7960:
7917:
7904:
7891:
7867:
7846:
7830:
7820:
7781:
7762:
7752:
7742:
7732:
7731:
7690:
7689:
7618:
7588:
7587:
7544:
7539:
7538:
7508:
7489:
7473:
7463:
7453:
7443:
7433:
7432:
7397:
7392:
7391:
7369:
7368:
7349:
7348:
7305:
7284:
7271:
7264:
7260:
7233:
7201:
7196:
7195:
7171:
7166:
7165:
7130:
7125:
7124:
7092:
7087:
7086:
7048:
7027:
7011:
7001:
6962:
6943:
6933:
6923:
6913:
6912:
6874:
6850:
6845:
6844:
6812:
6788:
6783:
6782:
6750:
6745:
6744:
6706:
6701:
6700:
6668:
6663:
6662:
6633:
6620:
6607:
6580:
6559:
6543:
6533:
6494:
6475:
6465:
6455:
6445:
6444:
6409:
6393:
6382:
6381:
6343:
6338:
6337:
6305:
6300:
6299:
6265:
6250:
6245:
6244:
6190:
6163:
6130:
6125:
6124:
6091:
6076:
6055:
6039:
6029:
6014:
5986:
5970:
5960:
5950:
5940:
5930:
5929:
5898:
5883:
5862:
5846:
5836:
5821:
5793:
5777:
5767:
5757:
5747:
5737:
5736:
5694:
5684:
5683:
5653:
5635:
5614:
5609:
5608:
5573:
5572:
5551:
5550:
5527:
5526:
5486:
5481:
5480:
5448:
5447:
5415:
5410:
5409:
5379:
5378:
5319:
5297:
5287:
5277:
5276:
5238:
5233:
5232:
5200:
5195:
5194:
5162:
5157:
5156:
5120:
5115:
5114:
5082:
5077:
5076:
5055:
5054:
5051:
5045:approximation.
4972:
4945:
4921:
4908:
4900:
4899:
4864:
4863:
4857:
4821:
4820:
4785:
4780:
4756:
4751:
4750:
4718:
4703:
4685:
4649:
4614:
4599:
4595:
4589:
4579:
4574:
4573:
4569:
4568:
4548:
4524:
4514:
4494:
4478:
4460:
4455:
4447:
4446:
4402:
4397:
4373:
4372:
4353:
4343:
4325:
4309:
4291:
4286:
4282:
4259:
4249:
4231:
4216:
4211:
4203:
4202:
4178:
4173:
4172:
4148:
4143:
4142:
4113:
4103:
4085:
4070:
4062:
4061:
4036:
4031:
4030:
3995:
3985:
3970:
3955:
3950:
3931:
3913:
3903:
3885:
3870:
3865:
3850:
3816:
3803:
3798:
3782:
3773:
3750:
3727:
3722:
3704:
3699:
3698:
3674:
3669:
3668:
3627:
3618:
3595:
3583:
3582:
3544:
3539:
3538:
3479:
3474:
3473:
3441:
3440:
3359:
3333:
3325:
3324:
3245:
3244:
3220:
3219:
3179:
3174:
3169:
3168:
3133:
3120:
3115:
3094:
3065:
3050:
3016:
3011:
2992:
2983:
2960:
2948:
2947:
2913:
2912:
2882:
2877:
2856:
2827:
2793:
2788:
2780:
2779:
2738:
2709:
2658:
2657:
2614:
2609:
2608:
2570:
2569:
2505:
2501:
2471:
2467:
2427:
2414:
2409:
2408:
2381:
2376:
2375:
2354:
2353:
2331:
2326:
2325:
2324:approximation,
2297:
2296:
2275:
2274:
2253:
2248:
2247:
2226:
2221:
2220:
2195:
2190:
2189:
2115:
2094:
2062:
2052:
2047:
2046:
2022:
2017:
2016:
1976:
1971:
1970:
1933:
1929:
1902:
1882:
1878:
1864:
1851:
1850:
1797:
1792:
1791:
1778:
1771:
1735:
1730:
1729:
1705:
1700:
1699:
1663:
1648:
1627:
1622:
1621:
1587:
1571:
1555:
1537:
1521:
1520:
1494:
1478:
1477:
1431:
1427:
1395:
1373:
1362:
1352:
1351:
1315:
1276:
1266:
1192:
1187:
1186:
1159:
1146:
1135:
1134:
1075:
1051:
1027:
1022:
1021:
1008:
919:
906:
898:
897:
888:
836:
831:
830:
820:
801:
743:
742:
696:
642:
637:
636:
629:The electronic
613:
602:
545:
506:
479:
474:
473:
430:
425:
424:
412:
375:
370:
369:
342:
337:
336:
294:
289:
288:
257:
252:
251:
209:
182:
177:
176:
152:
139:
126:
113:
100:
95:
94:
35:
28:
23:
22:
15:
12:
11:
5:
8528:
8526:
8518:
8517:
8512:
8507:
8505:Approximations
8502:
8492:
8491:
8486:
8485:
8479:
8474:
8469:
8458:
8457:External links
8455:
8453:
8452:
8409:
8366:
8322:
8279:
8272:
8246:
8237:(1954). "IV".
8222:
8209:
8203:978-0375726262
8202:
8184:
8140:
8112:
8110:
8107:
8104:
8103:
8093:
8080:
8068:
8051:
8040:
8029:
8022:
7999:
7998:
7996:
7993:
7992:
7991:
7986:
7981:
7976:
7971:
7966:
7959:
7956:
7941:
7940:
7929:
7924:
7920:
7916:
7911:
7907:
7903:
7898:
7894:
7890:
7887:
7884:
7881:
7876:
7871:
7866:
7863:
7860:
7855:
7850:
7845:
7842:
7836:
7833:
7827:
7823:
7817:
7814:
7811:
7808:
7805:
7802:
7797:
7790:
7787:
7780:
7777:
7774:
7769:
7765:
7758:
7755:
7749:
7745:
7739:
7716:
7713:
7709:
7704:
7700:
7697:
7686:
7685:
7674:
7671:
7668:
7664:
7659:
7655:
7652:
7649:
7646:
7643:
7639:
7634:
7630:
7625:
7621:
7617:
7614:
7611:
7607:
7602:
7598:
7595:
7572:
7569:
7565:
7560:
7556:
7551:
7547:
7535:
7534:
7523:
7520:
7515:
7511:
7507:
7504:
7501:
7496:
7492:
7488:
7485:
7480:
7476:
7470:
7466:
7459:
7456:
7450:
7446:
7440:
7417:
7413:
7409:
7404:
7400:
7376:
7356:
7345:
7344:
7333:
7329:
7325:
7322:
7318:
7313:
7309:
7304:
7300:
7296:
7292:
7288:
7283:
7278:
7274:
7270:
7267:
7263:
7259:
7256:
7253:
7249:
7245:
7240:
7236:
7232:
7229:
7226:
7222:
7217:
7213:
7208:
7204:
7178:
7174:
7150:
7146:
7142:
7137:
7133:
7112:
7108:
7104:
7099:
7095:
7083:
7082:
7071:
7068:
7065:
7062:
7057:
7052:
7047:
7044:
7041:
7036:
7031:
7026:
7023:
7017:
7014:
7008:
7004:
6998:
6995:
6992:
6989:
6986:
6983:
6978:
6971:
6968:
6961:
6958:
6955:
6950:
6946:
6939:
6936:
6930:
6926:
6920:
6900:
6897:
6894:
6890:
6886:
6881:
6877:
6873:
6870:
6866:
6862:
6857:
6853:
6832:
6828:
6824:
6819:
6815:
6811:
6808:
6804:
6800:
6795:
6791:
6770:
6766:
6762:
6757:
6753:
6732:
6729:
6726:
6722:
6718:
6713:
6709:
6688:
6684:
6680:
6675:
6671:
6659:
6648:
6645:
6640:
6636:
6632:
6627:
6623:
6619:
6614:
6610:
6606:
6603:
6600:
6597:
6594:
6589:
6584:
6579:
6576:
6573:
6568:
6563:
6558:
6555:
6549:
6546:
6540:
6536:
6530:
6527:
6524:
6521:
6518:
6515:
6510:
6503:
6500:
6493:
6490:
6487:
6482:
6478:
6471:
6468:
6462:
6458:
6452:
6433:
6432:
6421:
6416:
6412:
6408:
6405:
6400:
6396:
6392:
6389:
6363:
6359:
6355:
6350:
6346:
6325:
6321:
6317:
6312:
6308:
6287:
6283:
6279:
6274:
6269:
6264:
6259:
6254:
6226:
6221:
6217:
6213:
6210:
6207:
6203:
6197:
6193:
6189:
6186:
6183:
6179:
6175:
6170:
6166:
6162:
6159:
6155:
6151:
6146:
6139:
6136:
6121:
6120:
6109:
6106:
6103:
6098:
6094:
6090:
6085:
6080:
6075:
6072:
6069:
6064:
6059:
6054:
6051:
6045:
6042:
6036:
6032:
6026:
6021:
6017:
6013:
6010:
6007:
6002:
5995:
5992:
5985:
5982:
5977:
5973:
5967:
5963:
5956:
5953:
5947:
5943:
5937:
5927:
5916:
5913:
5910:
5905:
5901:
5897:
5892:
5887:
5882:
5879:
5876:
5871:
5866:
5861:
5858:
5852:
5849:
5843:
5839:
5833:
5828:
5824:
5820:
5817:
5814:
5809:
5802:
5799:
5792:
5789:
5784:
5780:
5774:
5770:
5763:
5760:
5754:
5750:
5744:
5706:
5701:
5697:
5692:
5680:
5679:
5668:
5665:
5660:
5656:
5652:
5648:
5642:
5638:
5634:
5631:
5626:
5623:
5618:
5593:
5589:
5585:
5581:
5559:
5534:
5506:
5502:
5498:
5493:
5489:
5468:
5464:
5460:
5456:
5435:
5431:
5427:
5422:
5418:
5397:
5393:
5389:
5386:
5375:
5374:
5363:
5360:
5357:
5354:
5351:
5348:
5345:
5341:
5337:
5334:
5331:
5326:
5322:
5318:
5315:
5312:
5309:
5303:
5300:
5294:
5290:
5284:
5258:
5254:
5250:
5245:
5241:
5220:
5216:
5212:
5207:
5203:
5182:
5178:
5174:
5169:
5165:
5140:
5136:
5132:
5127:
5123:
5102:
5098:
5094:
5089:
5085:
5063:
5050:
5047:
5035:
5034:
5023:
5020:
5017:
5014:
5011:
5008:
5005:
5002:
4992:
4988:
4984:
4979:
4975:
4971:
4968:
4965:
4961:
4957:
4952:
4948:
4944:
4941:
4937:
4933:
4928:
4924:
4920:
4911:
4907:
4881:
4876:
4872:
4855:
4838:
4833:
4829:
4805:
4801:
4797:
4791:
4788:
4783:
4779:
4776:
4772:
4768:
4763:
4759:
4744:
4743:
4732:
4727:
4722:
4717:
4712:
4707:
4702:
4697:
4694:
4689:
4675:
4671:
4667:
4662:
4656:
4652:
4647:
4640:
4635:
4632:
4628:
4621:
4617:
4611:
4608:
4603:
4598:
4592:
4585:
4582:
4577:
4572:
4565:
4561:
4555:
4551:
4547:
4544:
4539:
4535:
4531:
4527:
4521:
4517:
4512:
4508:
4502:
4497:
4493:
4488:
4485:
4481:
4477:
4473:
4466:
4463:
4458:
4454:
4440:
4439:
4428:
4422:
4418:
4414:
4408:
4405:
4400:
4396:
4393:
4389:
4385:
4380:
4376:
4368:
4364:
4360:
4356:
4350:
4346:
4341:
4337:
4328:
4324:
4319:
4316:
4312:
4308:
4304:
4297:
4294:
4289:
4285:
4279:
4274:
4270:
4266:
4262:
4256:
4252:
4247:
4241:
4238:
4234:
4229:
4222:
4219:
4214:
4210:
4185:
4181:
4158:
4155:
4151:
4128:
4124:
4120:
4116:
4110:
4106:
4101:
4095:
4092:
4088:
4083:
4077:
4073:
4069:
4049:
4046:
4042:
4039:
4029:The diagonal (
4027:
4026:
4015:
4010:
4006:
4002:
3998:
3992:
3988:
3983:
3973:
3968:
3961:
3958:
3953:
3949:
3946:
3941:
3938:
3934:
3928:
3924:
3920:
3916:
3910:
3906:
3901:
3895:
3892:
3888:
3883:
3876:
3873:
3868:
3864:
3857:
3853:
3849:
3842:
3839:
3836:
3832:
3828:
3819:
3813:
3809:
3806:
3801:
3797:
3792:
3788:
3785:
3778:
3772:
3768:
3764:
3754:
3747:
3742:
3737:
3733:
3730:
3725:
3720:
3716:
3707:
3677:
3637:
3633:
3630:
3623:
3617:
3613:
3609:
3599:
3592:
3566:
3562:
3558:
3548:
3526:
3523:
3520:
3517:
3514:
3511:
3508:
3502:
3499:
3495:
3491:
3486:
3482:
3461:
3457:
3453:
3449:
3437:
3436:
3425:
3422:
3418:
3414:
3410:
3406:
3403:
3400:
3396:
3392:
3388:
3384:
3381:
3377:
3373:
3363:
3358:
3355:
3351:
3347:
3337:
3332:
3314:
3313:
3302:
3298:
3294:
3290:
3286:
3283:
3280:
3277:
3274:
3270:
3266:
3262:
3258:
3255:
3252:
3228:
3207:
3203:
3199:
3195:
3191:
3185:
3182:
3177:
3165:
3164:
3153:
3149:
3145:
3140:
3136:
3130:
3126:
3123:
3118:
3114:
3109:
3105:
3101:
3097:
3093:
3089:
3085:
3081:
3077:
3072:
3068:
3063:
3053:
3048:
3044:
3040:
3036:
3032:
3028:
3022:
3019:
3014:
3010:
3007:
3002:
2998:
2995:
2988:
2982:
2978:
2974:
2964:
2957:
2928:
2924:
2920:
2909:
2908:
2897:
2892:
2888:
2885:
2880:
2876:
2871:
2867:
2863:
2859:
2855:
2851:
2847:
2843:
2839:
2834:
2830:
2825:
2821:
2817:
2813:
2809:
2805:
2799:
2796:
2791:
2787:
2773:
2772:
2761:
2758:
2754:
2750:
2745:
2741:
2737:
2733:
2729:
2725:
2721:
2716:
2712:
2706:
2701:
2698:
2695:
2691:
2687:
2684:
2680:
2676:
2672:
2668:
2665:
2642:
2638:
2634:
2630:
2626:
2621:
2617:
2596:
2592:
2588:
2584:
2580:
2577:
2563:
2562:
2551:
2548:
2545:
2542:
2539:
2536:
2533:
2530:
2515:
2512:
2508:
2504:
2499:
2495:
2491:
2487:
2483:
2478:
2474:
2470:
2464:
2461:
2458:
2455:
2451:
2447:
2443:
2439:
2434:
2430:
2424:
2421:
2417:
2388:
2384:
2362:
2338:
2334:
2305:
2283:
2260:
2256:
2233:
2229:
2202:
2198:
2186:
2185:
2174:
2171:
2168:
2165:
2162:
2159:
2156:
2153:
2143:
2139:
2135:
2131:
2127:
2122:
2118:
2114:
2110:
2106:
2101:
2097:
2093:
2090:
2086:
2082:
2078:
2074:
2069:
2065:
2055:
2025:
2004:
2000:
1996:
1992:
1988:
1983:
1979:
1963:
1962:
1951:
1943:
1940:
1936:
1932:
1928:
1923:
1920:
1917:
1912:
1909:
1905:
1889:
1885:
1881:
1874:
1871:
1867:
1861:
1858:
1854:
1845:
1842:
1839:
1836:
1833:
1830:
1827:
1823:
1817:
1813:
1809:
1800:
1776:
1769:
1745:
1742:
1738:
1715:
1712:
1708:
1678:
1672:
1667:
1662:
1657:
1652:
1646:
1642:
1637:
1634:
1630:
1618:inertial frame
1605:
1602:
1597:
1594:
1590:
1586:
1581:
1578:
1574:
1570:
1565:
1562:
1558:
1554:
1551:
1546:
1541:
1536:
1533:
1529:
1508:
1503:
1498:
1493:
1490:
1486:
1474:
1473:
1462:
1456:
1451:
1447:
1438:
1434:
1430:
1426:
1418:
1414:
1410:
1407:
1398:
1383:
1380:
1376:
1369:
1365:
1359:
1355:
1346:
1343:
1340:
1336:
1332:
1325:
1322:
1318:
1314:
1307:
1304:
1301:
1297:
1293:
1286:
1283:
1279:
1273:
1269:
1261:
1258:
1255:
1251:
1247:
1241:
1236:
1232:
1226:
1223:
1215:
1211:
1207:
1204:
1195:
1180:
1179:
1162:
1158:
1149:
1145:
1142:
1124:
1123:
1110:
1101:
1098:
1095:
1091:
1087:
1082:
1078:
1074:
1071:
1067:
1063:
1058:
1054:
1050:
1047:
1043:
1039:
1034:
1030:
1007:
1004:
988:
987:
976:
972:
968:
965:
962:
959:
956:
952:
948:
945:
942:
939:
935:
931:
922:
918:
909:
905:
886:
856:
852:
848:
843:
839:
825:. This is the
818:
814:, one obtains
799:
769:
765:
761:
757:
753:
750:
739:
738:
727:
723:
719:
715:
711:
708:
699:
695:
692:
688:
684:
680:
676:
673:
670:
666:
662:
658:
654:
645:
624:clamped-nuclei
611:
600:
594:kinetic energy
571:
568:
565:
562:
559:
556:
553:
548:
541:
538:
535:
532:
529:
526:
523:
520:
517:
514:
509:
505:
499:
496:
493:
490:
487:
482:
450:
447:
444:
441:
438:
433:
411:
408:
382:
378:
357:
354:
349:
345:
320:
316:
312:
309:
306:
301:
297:
276:
272:
269:
264:
260:
208:
205:
185:
164:
155:
151:
142:
138:
129:
125:
116:
112:
103:
59:wave functions
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
8527:
8516:
8513:
8511:
8508:
8506:
8503:
8501:
8498:
8497:
8495:
8488:
8483:
8480:
8478:
8475:
8473:
8470:
8467:
8464:
8463:
8462:
8456:
8448:
8444:
8440:
8436:
8432:
8428:
8424:
8420:
8413:
8410:
8405:
8401:
8397:
8393:
8389:
8385:
8381:
8377:
8370:
8367:
8362:
8358:
8354:
8350:
8346:
8342:
8338:
8334:
8326:
8323:
8318:
8314:
8310:
8306:
8302:
8298:
8294:
8290:
8283:
8280:
8275:
8269:
8265:
8261:
8257:
8250:
8247:
8242:
8241:
8236:
8232:
8226:
8223:
8219:
8213:
8210:
8205:
8199:
8195:
8188:
8185:
8179:
8174:
8170:
8166:
8162:
8159:(in German).
8158:
8154:
8147:
8145:
8141:
8128:
8124:
8117:
8114:
8108:
8097:
8094:
8090:
8084:
8081:
8078:
8072:
8069:
8065:
8061:
8057:
8050:
8046:
8039:
8035:
8028:
8021:
8017:
8014:
8010:
8004:
8001:
7994:
7990:
7987:
7985:
7982:
7980:
7977:
7975:
7972:
7970:
7967:
7965:
7962:
7961:
7957:
7955:
7953:
7950:
7944:
7927:
7922:
7918:
7909:
7905:
7901:
7896:
7892:
7885:
7882:
7874:
7869:
7861:
7853:
7848:
7843:
7834:
7831:
7825:
7815:
7812:
7809:
7803:
7800:
7795:
7785:
7775:
7772:
7767:
7756:
7753:
7747:
7737:
7730:
7729:
7728:
7695:
7672:
7650:
7647:
7623:
7619:
7615:
7593:
7586:
7585:
7584:
7549:
7545:
7521:
7518:
7513:
7509:
7502:
7499:
7494:
7490:
7483:
7478:
7474:
7468:
7457:
7454:
7448:
7438:
7431:
7430:
7429:
7402:
7398:
7390:The function
7388:
7331:
7327:
7311:
7298:
7294:
7290:
7281:
7272:
7268:
7265:
7261:
7257:
7254:
7238:
7234:
7230:
7206:
7202:
7194:
7193:
7192:
7176:
7172:
7162:
7135:
7131:
7097:
7093:
7069:
7066:
7063:
7055:
7050:
7042:
7034:
7029:
7024:
7015:
7012:
7006:
6996:
6993:
6990:
6984:
6981:
6976:
6966:
6956:
6953:
6948:
6937:
6934:
6928:
6918:
6898:
6895:
6879:
6875:
6871:
6855:
6851:
6817:
6813:
6809:
6793:
6789:
6755:
6751:
6730:
6727:
6711:
6707:
6673:
6669:
6660:
6646:
6643:
6638:
6634:
6625:
6621:
6617:
6612:
6608:
6601:
6598:
6595:
6587:
6582:
6574:
6566:
6561:
6556:
6547:
6544:
6538:
6528:
6525:
6522:
6516:
6513:
6508:
6498:
6488:
6485:
6480:
6469:
6466:
6460:
6450:
6442:
6438:
6437:
6436:
6419:
6414:
6410:
6406:
6403:
6398:
6394:
6390:
6387:
6380:
6379:
6378:
6375:
6348:
6344:
6310:
6306:
6272:
6267:
6262:
6257:
6252:
6243:= 1, 2), and
6242:
6224:
6219:
6215:
6208:
6205:
6201:
6195:
6184:
6168:
6164:
6160:
6144:
6134:
6107:
6104:
6101:
6096:
6092:
6083:
6078:
6070:
6062:
6057:
6052:
6043:
6040:
6034:
6024:
6019:
6015:
6008:
6005:
6000:
5990:
5980:
5975:
5971:
5965:
5954:
5951:
5945:
5935:
5928:
5914:
5911:
5908:
5903:
5899:
5890:
5885:
5877:
5869:
5864:
5859:
5850:
5847:
5841:
5831:
5826:
5822:
5815:
5812:
5807:
5797:
5787:
5782:
5778:
5772:
5761:
5758:
5752:
5742:
5735:
5734:
5733:
5730:
5728:
5724:
5723:Hilbert space
5720:
5699:
5695:
5666:
5658:
5654:
5640:
5636:
5629:
5624:
5621:
5616:
5607:
5606:
5605:
5579:
5548:
5524:
5520:
5491:
5487:
5420:
5416:
5361:
5358:
5355:
5346:
5343:
5332:
5324:
5316:
5313:
5301:
5298:
5292:
5282:
5275:
5274:
5273:
5270:
5243:
5239:
5205:
5201:
5167:
5163:
5154:
5125:
5121:
5087:
5083:
5048:
5046:
5044:
5038:
5021:
5018:
5015:
5012:
5009:
5006:
5003:
5000:
4977:
4973:
4969:
4966:
4950:
4946:
4926:
4922:
4918:
4909:
4898:
4897:
4896:
4879:
4874:
4870:
4861:
4854:
4836:
4831:
4827:
4817:
4789:
4786:
4781:
4777:
4761:
4757:
4747:
4730:
4725:
4715:
4710:
4700:
4695:
4692:
4660:
4654:
4650:
4645:
4638:
4633:
4630:
4626:
4619:
4609:
4606:
4590:
4583:
4580:
4575:
4570:
4563:
4559:
4553:
4549:
4545:
4542:
4519:
4515:
4495:
4491:
4486:
4483:
4479:
4464:
4461:
4456:
4445:
4444:
4443:
4426:
4406:
4403:
4398:
4394:
4378:
4374:
4348:
4344:
4326:
4322:
4317:
4314:
4310:
4295:
4292:
4287:
4277:
4254:
4250:
4239:
4236:
4232:
4220:
4217:
4212:
4201:
4200:
4199:
4183:
4179:
4156:
4153:
4149:
4108:
4104:
4093:
4090:
4086:
4075:
4071:
4047:
4044:
4040:
4037:
4013:
3990:
3986:
3971:
3959:
3956:
3951:
3944:
3939:
3936:
3932:
3908:
3904:
3893:
3890:
3886:
3874:
3871:
3866:
3855:
3851:
3847:
3840:
3837:
3834:
3830:
3826:
3817:
3811:
3807:
3804:
3799:
3795:
3790:
3786:
3783:
3740:
3735:
3731:
3728:
3705:
3697:
3696:
3695:
3675:
3666:
3660:
3657:
3651:
3635:
3631:
3628:
3580:
3537:. The matrix
3524:
3521:
3518:
3515:
3512:
3509:
3506:
3500:
3484:
3480:
3472:has elements
3423:
3404:
3401:
3356:
3323:
3322:
3321:
3319:
3292:
3278:
3275:
3264:
3250:
3243:
3242:
3241:
3197:
3183:
3180:
3175:
3138:
3134:
3128:
3124:
3121:
3116:
3112:
3083:
3070:
3066:
3051:
3034:
3020:
3017:
3012:
3005:
3000:
2996:
2993:
2946:
2945:
2944:
2942:
2895:
2890:
2886:
2883:
2878:
2874:
2845:
2832:
2828:
2811:
2797:
2794:
2789:
2778:
2777:
2776:
2759:
2743:
2739:
2727:
2714:
2710:
2704:
2699:
2696:
2693:
2689:
2685:
2674:
2656:
2655:
2654:
2632:
2619:
2615:
2586:
2566:
2549:
2546:
2543:
2540:
2537:
2534:
2531:
2528:
2513:
2510:
2506:
2489:
2476:
2472:
2462:
2459:
2456:
2445:
2432:
2428:
2422:
2419:
2415:
2407:
2406:
2405:
2402:
2386:
2382:
2336:
2332:
2323:
2318:
2258:
2254:
2231:
2227:
2218:
2200:
2196:
2172:
2169:
2166:
2163:
2160:
2157:
2154:
2151:
2133:
2120:
2116:
2099:
2095:
2091:
2080:
2067:
2063:
2053:
2045:
2044:
2043:
2023:
1994:
1981:
1977:
1968:
1949:
1941:
1938:
1934:
1921:
1918:
1915:
1910:
1907:
1903:
1887:
1883:
1879:
1872:
1869:
1865:
1859:
1856:
1852:
1843:
1840:
1837:
1834:
1831:
1828:
1825:
1821:
1815:
1811:
1807:
1798:
1790:
1789:
1788:
1785:
1783:
1779:
1772:
1765:
1759:
1743:
1740:
1736:
1713:
1710:
1706:
1697:
1693:
1670:
1660:
1655:
1640:
1635:
1632:
1628:
1619:
1595:
1592:
1588:
1584:
1579:
1576:
1572:
1568:
1563:
1560:
1556:
1549:
1544:
1531:
1501:
1488:
1460:
1454:
1449:
1436:
1432:
1428:
1424:
1416:
1412:
1408:
1405:
1396:
1381:
1378:
1374:
1367:
1363:
1357:
1353:
1344:
1341:
1338:
1334:
1330:
1323:
1320:
1316:
1312:
1305:
1302:
1299:
1295:
1291:
1284:
1281:
1277:
1271:
1267:
1259:
1256:
1253:
1249:
1245:
1239:
1234:
1224:
1221:
1213:
1209:
1205:
1202:
1193:
1185:
1184:
1183:
1160:
1156:
1147:
1143:
1140:
1133:
1132:
1131:
1129:
1099:
1096:
1080:
1076:
1072:
1056:
1052:
1048:
1032:
1028:
1020:
1019:
1018:
1015:
1013:
1005:
1003:
1001:
997:
993:
963:
960:
957:
943:
920:
916:
907:
896:
895:
894:
892:
885:
880:
878:
874:
870:
841:
837:
828:
824:
817:
813:
809:
805:
798:
795:
791:
787:
783:
759:
748:
717:
706:
697:
693:
682:
671:
660:
643:
635:
634:
633:
632:
627:
625:
621:
617:
610:
606:
599:
595:
590:
588:
546:
507:
503:
471:
467:
422:
417:
409:
407:
405:
400:
398:
380:
376:
355:
352:
347:
343:
334:
318:
314:
310:
307:
304:
299:
295:
274:
270:
267:
262:
258:
248:
246:
241:
237:
232:
230:
226:
222:
221:energy levels
218:
214:
206:
204:
183:
162:
153:
149:
140:
136:
127:
123:
114:
110:
101:
92:
88:
85:In molecular
83:
79:
77:
73:
68:
64:
63:atomic nuclei
60:
56:
55:approximation
52:
48:
44:
40:
33:
19:
8487:
8460:
8422:
8418:
8412:
8379:
8375:
8369:
8336:
8332:
8325:
8292:
8288:
8282:
8255:
8249:
8238:
8225:
8217:
8212:
8193:
8187:
8160:
8156:
8131:. Retrieved
8126:
8116:
8096:
8083:
8071:
8063:
8059:
8048:
8044:
8037:
8033:
8026:
8019:
8015:
8003:
7989:Kohn anomaly
7945:
7942:
7687:
7536:
7389:
7346:
7163:
7084:
6440:
6434:
6376:
6240:
6122:
5731:
5681:
5522:
5518:
5376:
5271:
5152:
5052:
5039:
5036:
4859:
4852:
4818:
4748:
4745:
4441:
4028:
3665:Leibniz rule
3661:
3652:
3578:
3438:
3317:
3315:
3166:
2910:
2774:
2567:
2564:
2403:
2319:
2216:
2187:
1966:
1964:
1786:
1781:
1774:
1767:
1764:atomic units
1760:
1695:
1694:and nucleus
1691:
1475:
1181:
1127:
1125:
1016:
1009:
995:
989:
890:
883:
881:
876:
872:
822:
815:
807:
803:
796:
789:
785:
781:
740:
628:
623:
615:
608:
597:
591:
421:wavefunction
413:
401:
332:
249:
235:
233:
210:
158:nuclear spin
90:
89:, using the
87:spectroscopy
84:
80:
54:
50:
46:
36:
8468:(in German)
8101:dependence.
7949:Jahn–Teller
466:vibrational
132:vibrational
8494:Categories
8109:References
1006:Derivation
794:eigenvalue
470:rotational
240:eigenvalue
236:electronic
188:electronic
145:rotational
119:electronic
8447:0009-2614
8404:0021-9606
8361:0021-9606
8317:0009-2614
8235:Huang, K.
8009:electrons
7919:χ
7902:−
7883:η
7870:τ
7865:∇
7859:∇
7849:τ
7822:ℏ
7810:η
7801:−
7789:~
7773:η
7764:∇
7744:ℏ
7738:−
7712:Γ
7696:η
7667:Γ
7651:η
7642:Γ
7620:χ
7610:Γ
7594:χ
7568:Γ
7546:χ
7510:ξ
7500:−
7475:ξ
7465:∇
7445:ℏ
7439:−
7399:ξ
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7355:Γ
7321:Γ
7299:τ
7295:⋅
7277:Γ
7273:∫
7266:−
7258:
7235:ξ
7225:Γ
7203:χ
7173:χ
7064:χ
7051:τ
7046:∇
7040:∇
7030:τ
7003:ℏ
6991:χ
6982:−
6970:~
6954:χ
6945:∇
6925:ℏ
6919:−
6896:∼
6872:−
6810:∼
6728:∼
6708:ψ
6635:ψ
6618:−
6596:χ
6583:τ
6578:∇
6572:∇
6562:τ
6535:ℏ
6523:χ
6514:−
6502:~
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6477:∇
6457:ℏ
6451:−
6411:ψ
6395:ψ
6388:χ
6268:τ
6253:τ
6216:τ
6192:ℏ
6138:~
6093:ψ
6079:τ
6074:∇
6068:∇
6058:τ
6031:ℏ
6016:ψ
6006:−
5994:~
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5942:ℏ
5936:−
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5875:∇
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5838:ℏ
5832:−
5823:ψ
5813:−
5801:~
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5749:ℏ
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5664:⟩
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5633:⟨
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5580:τ
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5353:Ψ
5344:−
5330:Ψ
5317:τ
5311:∇
5289:ℏ
5283:−
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4832:α
4778:≈
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4620:α
4576:χ
4560:∑
4526:⟩
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4395:−
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4318:α
4288:χ
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4261:⟩
4251:χ
4240:α
4213:χ
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4105:χ
4094:α
4072:χ
4068:⟨
3997:⟩
3987:χ
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3915:⟩
3905:χ
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3867:χ
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3841:α
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3827:−
3800:δ
3741:≡
3519:…
3481:ϕ
3448:ϕ
3409:ϕ
3387:ϕ
3282:Ψ
3254:Ψ
3176:χ
3117:δ
3096:⟩
3067:χ
3013:χ
3009:⟨
3006:≡
2879:δ
2858:⟩
2829:χ
2790:χ
2786:⟨
2740:ϕ
2711:χ
2690:∑
2664:Ψ
2616:χ
2576:Ψ
2529:α
2514:α
2503:∂
2473:χ
2469:∂
2460:−
2429:χ
2423:α
2383:χ
2333:χ
2322:(LCAO-MO)
2255:χ
2228:χ
2197:χ
2164:…
2117:χ
2064:χ
1978:χ
1942:α
1931:∂
1927:∂
1919:−
1911:α
1873:α
1860:α
1826:α
1822:∑
1812:∑
1661:−
1641:≡
1532:≡
1489:≡
1446:∇
1413:∑
1409:−
1335:∑
1296:∑
1250:∑
1246:−
1231:∇
1210:∑
1206:−
1100:⋯
1097:≪
1073:≪
1049:≪
964:ϕ
944:ϕ
749:χ
707:χ
672:χ
547:ψ
508:ψ
481:Ψ
432:Ψ
67:electrons
8510:Max Born
8231:Born, M.
8133:2 August
8032:implies
8013:momentum
7958:See also
7312:′
7291:′
5547:gradient
5043:diabatic
4790:′
4661:⟩
4584:′
4571:⟨
4465:′
4407:′
4296:′
4221:′
4041:′
3960:′
3875:′
3808:′
3787:′
3732:′
3656:diabatic
3632:′
3581:) terms
3184:′
3125:′
3021:′
2997:′
2887:′
2798:′
416:electron
72:Max Born
8427:Bibcode
8384:Bibcode
8341:Bibcode
8297:Bibcode
8165:Bibcode
7537:Having
5545:is the
829:(PES):
213:benzene
207:Example
8445:
8402:
8359:
8315:
8270:
8200:
7688:where
7347:where
6123:where
5571:, and
5377:where
3504:
741:where
331:(with
45:, the
7995:Notes
6843:, or
5682:Here
4858:(the
2775:with
1182:with
1128:exact
106:total
8443:ISSN
8400:ISSN
8357:ISSN
8313:ISSN
8268:ISBN
8198:ISBN
8135:2022
8043:) ≪
6336:and
5231:and
5113:and
4681:with
1898:with
1773:and
1728:and
1342:>
1303:>
356:1296
211:The
65:and
41:and
8435:doi
8423:265
8392:doi
8380:111
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8337:105
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8260:doi
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2524:for
2147:for
2015:of
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616:not
315:876
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790:R
786:r
782:R
768:)
764:R
760:,
756:r
752:(
726:)
722:R
718:,
714:r
710:(
702:e
698:E
694:=
691:)
687:R
683:,
679:r
675:(
669:)
665:R
661:,
657:r
653:(
648:e
644:H
612:e
609:H
601:n
598:T
570:r
567:a
564:e
561:l
558:c
555:u
552:n
540:c
537:i
534:n
531:o
528:r
525:t
522:c
519:e
516:l
513:e
504:=
498:l
495:a
492:t
489:o
486:t
449:l
446:a
443:t
440:o
437:t
423:(
381:2
377:n
353:=
348:2
333:N
319:N
308:=
305:N
300:2
268:=
263:2
184:E
163:.
154:E
150:+
141:E
137:+
128:E
124:+
115:E
111:=
102:E
49:(
34:.
20:)
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