Molecular Relativistic Corrections Determined in the Framework

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Molecular Relativistic Corrections Determined in the Framework where the Born-Oppenheimer Approximation is not Assumed Monika Stanke, and Ludwik Adamowicz J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/jp4020492 • Publication Date (Web): 16 May 2013 Downloaded from http://pubs.acs.org on May 23, 2013

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The Journal of Physical Chemistry

Molecular Relativistic Corrections Determined in the Framework where the Born-Oppenheimer Approximation is not Assumed. Monika Stanke∗,† and Ludwik Adamowicz∗,‡ Institute of Physics,Faculty of Physics, Astronomy and Informatics ul. Grudzia¸dzka 5, Toru´n, PL 87-100, Poland , and University of Arizona, Tucson, Arizona 85721, USA E-mail: [email protected]; [email protected]

∗ To

whom correspondence should be addressed Copernicus University ‡ Department of Chemistry and Biochemistry † Nicolaus

1 ACS Paragon Plus Environment


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Abstract In this work we describe how the energies obtained in molecular calculations performed without assuming the Born-Oppenheimer (BO) approximation can be augmented with corrections accounting for the leading relativistic effects. Unlike the conventional BO approach, where these effects only concern the relativistic interactions between the electrons, the non-BO approach also accounts for the relativistic effects due to the nuclei and due to the coupling of the coupled electron-nucleus motion. In the numerical sections the results obtained with the two approaches are compared. The first comparison concerns the the dissociation energies of the two-electron isotopologues of the H2 molecule, H2 , HD, D2 , T2 , and the HeH+ ion. The comparison shows that, as expected, the differences in the relativistic contributions obtained with the two approaches increase as the nuclei become lighter. The second comparison concerns the relativistic corrections to all 23 pure vibrational states of the HD+ ion. An interesting charge asymmetry caused by the non-adiabatic electron-nucleus interaction appears in this system and this effect significantly increases with the vibration excitation. The comparison of the non-BO results with the results obtained with the conventional BO approach, which in the lowest order does not describe the charge-asymmetry effect, reveals how this effect affects the values of the relativistic corrections.

Keywords: Non-adiabatic calculations, diatomic molecules, leading relativistic energy corrections, isotopologues of H2 , HD+ ion.

Introduction Recent progress in the development of high-precision laser-based spectroscopy techniques to study atomic and molecular spectra has been offering new capabilities to measure energy levels of small atoms and molecules in the gas phase with unprecedented accuracy. Due to these developments the energy levels of atomic and molecular bound states and the corresponding transition energies are known with sub 0.01 cm−1 accuracy.


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The experimental results can be directly compared with the theoretical calculations. This comparison not only tests the correctness of the quantum-mechanical model used in the calculations, but also it sheds light on the nature of the chemical bonding, contributes to the measurements of the fundamental constances, and provides information on the chemical composition of distant interstellar objects. Very accurate experimental measurements challenge the theoreticians to improve the quantum mechanical models used in the calculations and to carry out the development of new algorithms for more accurate determination of bound states of atomic and molecular systems. The measurements also provide date for testing more approximate quantum mechanical methods developed for the calculations of larger atoms and molecules. The increased accuracy of the theoretical models is accomplished by including new effects neglected in the previous models, which describe higher-order nuances originating from the quantum theory of interparticle interactions. To fully describe these interactions in molecules one needs to use the relativistic field theory which provides complete account of all relevant effects present in stationary bound states of these systems. One way to analytically describe a bound state of a quantum system with including the relativistic effects is to start with the non-relativistic formalism in the zeroth order and to use the perturbation theory to incorporate the effects due to relativistic interactions. In this way the quantum electrodynamics (QED) theory is replaced by a theory which was first formulated by Caswell and Lepage 1 and referred to as the nonrelativistic quantum electrodynamics (NRQED) theory (very lucid presentation of NRQED was given by Kinoshita 2 ). NRQED enables construction of an effective Hamiltonian, H eff , which replaces the QED Hamiltonian, HQED . This construction for bound states of light molecular quantum systems involves systematic inclusion of operators representing various relativistic and QED effects. The operators are usually expressed as terms of certain orders of the hyperfine constant α . Due to that, all effects originating from QED can be determined with the use of the Rayleigh-Schrödinger perturbation theory. 1 1 NRQED separates the contributions from the relativistic and nonrelativistic moments and splits them into terms proportional to consecutive powers of α . This type of separation appear neither in the Bethe-Salpeter (BS) analysis nor in the theory which employs arbitrary ”quasipotentials". The interaction center in the BS approach is constructed using the Feynman diagrams, which consist of linear moments of different orders in α (α 2 m, α 4 m, · · ·). In effect, this center comprises infinite number of momenta with different weights. The NRQED theory is an approximation which only includes nonrelativistic momenta. The NRQED Lagrangian is constructed from two-component Pauli spinors



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In this work we will describe the application of the NRQED theory in calculations of stationary bound states of diatomic molecules. We will particularly focus on calculating the relativistic corrections using non-relativistic all-particle wave functions obtained for small diatomics without assuming the Born-Oppenheimer (BO) approximation. To our knowledge ours have been the first calculations of this kind ever performed. The procedure for generating the non-BO nonrelativistic wave functions involves the internal non-relativistic Hamiltonian of the system obtained by rigorously separating the center-of-mass motion from the laboratory-frame all-particle Hamiltonian. This internal Hamiltonian is the unperturbed (zero-order) operator in our approach and it is used together with the operators representing higher-order effects in α , which are treated as perturbations, in the framework of the first-order Rayleigh-Schrödinger perturbation theory to calculate relativistic contributions to the energies of bound states of some small molecules.

Bound states of molecules with few electrons. In should be noted that the nonrelativistic formalism (Enr is used here to denote the nonrelativistic energy) is incomplete, but mathematically simple, and it can produce very accurate results for atoms and molecule containing small nucleus/nuclei. The accuracy achieved in the nonrelativistic description of atomic and molecular spectra does not match the accuracy which is now possible is the measurements involving modern experimental techniques used to characterize these spectra. The accuracy of the nonrelativistic calculations is also too low to assist in a qualitatively correct analysis of the spectra and in their assignment. All electrodynamic corrections to bound-state energies of the system derived from QED can be represented as terms proportional to powers of three fundamental quantities: 3 α , (α Z), and

m M.

This leads to the following: • Corrections which may be derived from the Dirac–Coulomb (DC) equation by expanding (i.e. nonrelativistic spinors) and must be invariant in terms of the gauge symmetry, the parity conservation, and the time inversion. It also must satisfy the locality requirement, the hermicity, and the invariance in terms of the Galileo transformation.


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the DC energy in a power series of (Z α ). The lowest-order corrections are proportional to (Z α )2 and are referred to as the Pauli corrections. Higher-order terms of the expansion are proportional to even powers of (Z α ). For example, the corrections proportional to (Z α )4 were recently derived. 4,5 • The energy corrections, which dependent on powers of α and Z α and which do not result from the DC, equation are called radiative corrections. The lowest-order corrections of this type are proportional to α 3 . • The corrections proportional to the ratios of masses m/M of the light and heavy particles. These corrections, referred to as the recoil corrections, provide a contribution to the energy when the finite nuclear masses are taken into account. • The last class of the corrections originate from the coupling between electromagnetic and weak interactions and from strong (nuclear) interactions. In this work the total energy of the system is represented as an expansion in terms of powers of

α: E = E (0) + α 2 E (2) + α 3 E (3) + · · · ,

(1)

where E (0) = Enr is the nonrelativistic energy, α 2 E (2) are the relativistic Pauli corrections, and

α 3 E (3) + · · · are QED corrections and higher order relativistic corrections. Only the α 2 E (2) corrections are considered in this work.

Hamiltonian The majority of atomic and molecular nonrelativistic calculations are performed assuming the Born-Oppenheimer (BO) approximation. Subsequently, the accuracy of the results (energies) are enhanced by including adiabatic and nonadiabatic corrections. A more direct and potentially more accurate and more rigorous (and perhaps also more interesting) approach is to develop methods



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where all the particles forming the system are described on equal footing from the beginning, i.e. without neglecting the coupling of the motions of the light particles (electrons, positrons) and heavy particles (nuclei). A representation of the system one gets from such an approach does not formally differentiate between the electrons and the nuclei, because the BO approximation is not assumed. A more pragmatic reason for considering a non-BO approach is the strive to obtain the most accurate quantum-mechanical representation of the system possible with the error in system’s total energy not exceeding a µ hartree. The strive is primarily motivated by the constantly increasing accuracy of the experimental techniques used to measure the atomic and molecular spectra. These measurements are starting to reach accuracy of sub 0.0001 cm−1 which is very hard to achieve in theoretical calculations based on first principles. The method for very accurate atomic and molecular calculations developed by the Adamowicz group in recent years 6–13 is based on equal treatment of all N particles forming the system, i.e. the electrons and the nuclei. The approach starts with the laboratory-frame Hamiltonian describing the motion and the interaction of the particles in a laboratory Cartesian coordinate system and then involves a transformation to new coordinates, first three of them being the lab coordinates of the center of mass and the remaining 3N − 3 being internal coordinates. The transformation allows for separating the total energy operator into a Hamiltonian representing the kinetic energy of the center-of-mass motion and a Hamiltonian representing the internal energy of the system. To describe a system consisting of N = n + 1 quantum particles with masses Mi and charges Qi (i = 1, . . ., n + 1) in the laboratory Cartesian coordinate system one needs to introduce n + 1 position vectors Ri and n + 1 momentum vectors Pi : These vectors are:







X1

 R1       Y1  R2      R = = Z    1  ···   .     ..  Rn+1 Zn+1









 P1        P2      P = =     ···        Pn+1

     ,      6

ACS Paragon Plus Environment

Px1 Py1 Pz1 .. . Pz(n+1) .

           

(2)

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The nonrelativistic Hamiltonian of the system is:

H nr (R) =

n+1

n+1 n+1 Qi Q j P2i + ∑ 2Mi ∑ ∑

Ri − R j

. i=1 i=1 j>i

(3)

Next, the 3(n + 1)-dimensional problem represented by the above Hamiltonian is reduced to a [3(n + 1) − 3]-dimensional problem by eliminating from the laboratory-frame Hamiltonian the center-of-mass motion. A transformation which allows to do that is:

r0 =

M(n+1) M2 M1 R1 + R2 + · · · + Rn+1 , Mtot Mtot Mtot

r1 = R2 − R1 r2 = R3 − R1 , ··· rn = Rn+1 − R1 , where Mtot = ∑N i=1 Mi . As one can see, the new coordinates system consists of the three Cartesian coordinates of the center of mass in the laboratory coordinate system, r0 , and 3n internal Cartesian coordinates, ri , describing the positions of particles 2 to n+1 relative to the position of particle 1, which is assumed to be the reference particle. The above transformation can be expressed in the following compact form:

T : R −→ r

r = TR

T : P −→ p

p = (T )−1 P,



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where transformation T is given as: 

M1 Mtot

   −1   T =  −1   ..  .  −1

M2 Mtot

M3 Mtot

···

1

0

···

0 .. .

1 .. .

··· .. .

0

0

···

MN Mtot



  0    0   ⊗ I3 ,  ..  .   1

(4)

R −→ [rTcm , rT ]T . By applying the coordinate transformation T to the laboratory-frame total Hamiltonian and by separating out the operator representing the center of mass motion the following internal Hamiltonian is obtained:

nr Htot (r0 , r) =

1 1 2 − ∇ 2 Mtot r0

+

n n qq 1 n 1 2 1 n 1 q0 qi i j − ∑ ∇ri − ∑ +∑ ∇ri · ∇ri + ∑ 2 i µi 2 i6= j m0 i< j ri j i=1 ri

where qi = Qi+1 denotes pseudoparticle charges and µi =

m0 mi m0 +mi

!

,

their reduced masses, m0 is the

mass of the reference particle, mi = Mi+1 , ∇ri is the gradinet vector expressed in terms of the x, y, z

coordinates of vector ri , ri j = ri − r j = Ri+1 − R j+1 , and r0i ≡ ri = kri k = kRi+1 − R1 k.

We call the particles described by the above Hamiltonian ”pseudoparticles" because, even though they have the same charges as the original particles, their masses are not the original masses but the reduced masses. The separation of the total nonrelativistic lab-frame Hamiltonian into the nr (r ), and the internal operator representing the kinetic energy of the center-of-mass motion, Hcm 0 nr (r), is rigorous: Hamiltonian, Hint

nr nr nr Htot (r0 , r) = Hcm (r0 ) + Hint (r).

(5)

nr (r ) and H nr (r) provides a complete description of the positions of all particles in The sum of Hcm 0 int

space. As in this work we are concerned with the internal bound states of the system, the solutions


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of the Schrödinger equation with the internal Hamiltonian will only be considered. This internal Hamiltonian can be viewed as describing a system of n pseudoparticles with the masses equal to reduced masses, µi , and charges qi (i = 1, . . . , n). The pseudoparticles interact with each other by the Coulombic potential and their motions are coupled through the mass-polarization terms. What is accomplished by the above-discussed coordinate transformation for an isolated system in vacuum, is also the separation of the energy into a non-quantized translational energy of the center-of-mass motion and the internal energy of the system, which for the bound states is quantized. Also, in the new coordinate system the total wave function of the system can be written as a simple product of a wave function for the center of mass and the wave function representing an internal state of the system:

Φ(r0 , r1 , r2 , · · ·) = Ψ(r0 ) Ψ(r1 , r2 , · · ·).

(6)

As one notices, the motion of the pseudoparticles, as described by the internal Hamiltonian, is a motion in the central filed of the charge of the reference particle placed in the center of the internal coordinate system. The charges of the pseudoparticles are the same as the charges of the original particles, i.e. the charge of the first pseudoparticle is equal to the charge of the original particle 2, the charge of the second pseudoparticle is equal to the charge of the original particle 3, nr as a generalized atomic Hamiltonian with an immobilized nucleus in etc. Thus, one can view Hint

the center of the coordinate system with the charge q0 and with electrons replaced by pseudoparticles. However, unlike the electrons, some of these pseudoparticles may have positive charges, if a molecular system is considered. For the hydrogen atom, the internal Hamiltonian describes the motion of a pseudoelectron with the reduced mass equal to µ =

me m p me +m p

(me and m p are the electron

and proton masses, respectively) in the central potential of the charge of the proton. As the internal Hamiltonian is fully symmetric (isotropic) with respect to all rotations around the center of the internal coordinate system, its wave functions transform as irreducible representations of the fully symmetric group of rotations.



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Non-Born-Oppenheimer calculations Correlated Gaussian Basis Set Not assuming the Born-Oppenheimer (BO) approximation and treating all particles in the system on equal footing involves the need to describe in the calculation the correlation effects due to different types of particles. In a molecular calculation this means describing the electron-electron (e-e), electron-nucleus (e-n), and nucleus-nucleus (n-n) correlation effects. To do this most effectively one needs to use basis functions which explicitly depend on the e-e, e-n, and n-n distances. As electrons are light in comparison to the nuclei, their ”individual" wave functions strongly overlap and the probability of two of them be found in one point in space (if they have opposite spins) is much greater than for two nuclei. This is because the nuclei are heavier to they avoid each other much more in their relative motion in the molecule. Also, the electrons, particularly the core electrons, follow very closely the motion of the nuclei because they are attracted to them via the Coulombic forces. These are the physical reasons leading to the correlation effects, which are the strongest for n-n and the weakest for e-e. The need to fulfill the symmetry requirements for the wave function and to very effectively describe the the correlation effects in the calculations turned our attention to explicitly correlated all-particle Gaussian (ECG) functions as the basis functions for atomic and molecular non-BO calculations. Nonrelativistic atomic non-BO calculations are performed with the internal Hamiltonian which describes the motion of pseudoelectrons around the center of the coordinate system, where the charge of the nucleus is located. For an atom with only s electrons the ECG basis functions used in the wave function expansion have the following form:

¯k r , ψk (r) = exp −rT A

(7)

¯ is the matrix of the variational exponential parameters. A ¯ is 3n-dimensional, symmetric, where A ¯ = A ⊗ I3 , and positive definite (three dimensions for each pseudoelectron). It can be written as: A


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where I3 is the 3 × 3 unit matrix. Function (7) is invariant upon any orthogonal unitary transformation, U , which is evident from: ((In ⊗U ) r)T (A ⊗ I3 ) (In ⊗U ) r = rT In ⊗U T (A ⊗ I3 ) (In ⊗U ) r = = rT A ⊗U T U r = rT (A ⊗ I3 ) r.

(8)

nr (r), has the Let us now consider a diatomic system. The diatomic internal Hamiltonian, Hint

same form as the internal atomic Hamiltonian (only one of the pseudoparticle now has a positive charge and is heavier than the pseudoelectrons) and it is spherically symmetric. This Hamiltonian commutes with the square of the total angular momentum of the system. Thus it is convenient to expand a system wave function using basis functions, which provide a basis set for the representation of the fully symmetric group of rotations the wave function belongs to, because with that states corresponding to different total-angular-momentum quantum numbers can be separated. Let us now consider only states with the zero total-angular-momentum quantum number (so-called pure vibrational states; we should note that, as the vibrational quantum number is not strictly speaking a good quantum number, the non-BO wave functions of the pure vibrational states corresponding to the ground electronic state contain some very small contributions of excited electronic states multiplied by their corresponding vibrational states). The calculations performed in this work concern only such states. The pure vibrational states can be described by fully spherically symmetric basis functions like (7). However, due to the need to more effectively describe the n-n correlation effects in the calculation for a diatomic system the atomic ECG basis set has to be modified. As the correlation effect increases with the increase of the masses of the correlating particles, the simple ECGs, which only depend on the n-n distance in the exponent, are not sufficient. Also, for each state of the molecule, which is in the electronic and rotational ground states, there is a certain distance (called equilibrium distance) at which the probability of finding the two nuclei is at maximum. As this distance is the same regardless which direction away from the center of the internal coordinate system one moves (the wave function is spherically symmetric), the maximum



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of wave function has a maximum on a sphere. To describe this behavior, not only in the ground vibrational states, but also in vibrationally excited states, whose wave function have radial nodes, one needs to multiply the simple atomic ECGs by powers of the internuclear distance, r1 : φk = r1mk exp −rT (Ak ⊗ I3 )r = r1mk exp −rT Ak r ,

(9)

where r1 = |R1 − R2 |. The factor r1mk shifts the maximum of the Gaussian away from the reference particle (which is located in the center of the coordinate system). The higher the power the larger the shift. mk has a value in the 0 − 250 range in our calculations.

Transformation of the relativistic operators to the internal coordinate system. The transformation of the nonrelativistic Hamiltonian from the laboratory coordinate system to the internal coordinate system and the separation of the center–of–mass motion was described in one of the previous sections. Now the transformation needs to be applied to the relativistic Hamiltonian (see, for example, Refs. 14,15 ). The Breit-Pauli relativistic Hamiltonian is used in this work. Separating out the center–of–mass motion reduces the (n + 1)-particle problem to a n-pseudo-particle problem. The positions of the pseudoparticles are described by vectors ri = Ri+1 − R1 . While a full separation of the laboratory non-relativistic Hamiltonian into the Hamiltonian describing the kinetic energy of the center–of–mass motion and the internal Hamiltonian is rigorous, the separation of the relativistic Hamiltonian into the internal and external parts is not. The Breit–Pauli Hamiltonian, after it is transformed to the new coordinate system (r, r0 ), can be written as a sum of three terms: rel rel rel H rel (r, r0 ) = Hcm (r0 ) + Hint (r) + Hcm−int (r, r0 ),

(10)

rel (r ) is the term describing the relativistic effects of the motion of the center of mass, where Hcm 0 rel (r) describes the internal relativistic effects, and H rel Hint cm−int (r, r0 ) describes the relativistic cou-

pling of the internal and external motions. This latter effect has not been considered in our calculations, as we assume that the system as a whole is at rest, i.e., the center of mass is not moving. 12 ACS Paragon Plus Environment

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rel . Thus the relativistic corrections to the internal bound states of the system are calculated using Hint

For a diatomic system for states with the S symmetry (J = 0), the transformation of the coordinate system leads to the relativistic mass-velocity (mv), one- and two-particle Darwin (d1 and d2), orbit-orbit (oo), and spin-spin (ss) operators in the following form (in the case of a system consisting of fermions and bosons, the reference particle is usually a boson with parameters: m0 , q0 , s0 ). The fermion terms are denoted as f − f and fermion-boson terms are denoted as f − b. Note, that using one of the nucleus of the diatomic system as the reference particle and after separating the center of mass, there is only one particle (pseudoparticle) in the system, which can be a boson. This is why there are no b − b terms in the relativistic Hamiltonians shown below.

f−f f −b Hmv = Hmv = −

f−f

f−f

= −

f −b

f −b

= −

Hd1 + Hd2

Hd1 + Hd2

f−f f −b Hoo + Hoo = −

− + Hssf − f = − − Hssf −b = −



!4

n

n



1 1 1 ∇ri + ∑ 3 ∇4ri  , ∑ 3 8 m0 i=1 i=1 mi " # n n n qi q j 3 π n q0 qi 3 q0 qi 3 ∑ m2 δ (ri) + ∑ m2 δ (ri) + ∑ ∑ m2 δ (ri j ) , 2 i=1 i i i=1 i=1 j=1, j6=i 0 " # n n qi q j 3 π n q0 qi 3 δ δ (ri j ) , (r ) + i ∑ ∑ ∑ 2 2 2 i=1 mi i=1 j=1, j6=i mi 1 1 n q0 qi 1 ∑ m0mi ri ∇ri · ∇ri + r3 ri · (ri · ∇ri )∇ri + 2 i=1 i n n 1 q0 qi 1 1 ∑ ∑ m0 mi ri ∇ri · ∇r j + r3 ri · (ri · ∇ri )∇r j + 2 i=1 j=1, j6=i i " # 1 1 n−1 n qi q j 1 ∑ ∑ mim j ri j ∇ri · ∇r j + r3 ri j · (ri j · ∇ri )∇r j 2 i=1 j>i ij 8π n q0 qi ∑ m0 mi (S0 · Si) δ 3 (ri) + 3 i=1 8π 3 8π 3

n

n

qi q j

∑ ∑ mim j (Si · S j ) δ 3 (ri j ),

j=1 i> j n n

qi q j

∑ ∑ mim j (Si · S j ) δ 3 (ri j ),

j=1 i> j

where, for consistency of the notation, we use mi = Mi+1 . We should note that the spin–orbit f−f

f−f

term, Hso , and the two terms describing the spin–spin interactions in Hss


vanish. 16 The only


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f−f

spin-dependent term which remains in Hss

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is the Fermi contact term.

Numerical examples So far our molecular non-BO calculations have concerned diatomic ground and excited states corresponding to the zero total rotational quantum number, J = 0. As in the non-BO approach the motions of the electrons and the nuclei are treated on equal footing, the electronic and nuclear states mix. However, the states corresponding to the motion of the nuclei becoming excited are lower in energy and much more closely spaced in the spectrum than the states corresponding to the electronic excitations. This happens due to the difference in the masses of these two types of particles. Therefore, when one calculates the spectrum of states corresponding to J = 0, the lower part of the spectrum consists of states where only the nuclear motion becomes excited. Such states differ from each other in terms of the number of nodes in terms of internuclear coordinates. For a diatomic molecule, in the internal coordinate system used in our non-BO calculations, such a coordinate is the distance between the first and the second nucleus denoted as r1 . Thus, if the basis set used in the calculation is restricted to functions corresponding to J = 0, the calculation yields states, which are commonly called pure vibrational states, although, strictly speaking, the vibrational quantum number is only approximately a good quantum number for molecular states. This is because, as higher states are considered in the calculation employing J = 0 basis functions, lowerlying electronic excited states may start to provide (small) contributions to the wave functions of those states. A good example of such a situation is the HD+ ion, which will be discussed next, where for top two states obtained in non-BO calculations performed with J = 0 basis functions, the electronic Σu excited state mixes in with the Σg ground state resulting in a significant charge asymmetry. The numerical results concerning the pure vibrational states obtained in our non-BO calculations, which will be discussed next, are compared with the results obtained by others using the conventional approach. In that approach one first calculates the electronic nonrelativistic electronic


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energies and the corresponding wave functions on a grid of points representing different geometries of the molecule under consideration. For a diatomic molecules the calculations are done for different internuclear distances ranging from short to long. In this way the potential energy curve (PEC), or the potential energy surface (PES) for a molecule with more than two nuclei, is generated. If the calculation are to provide very accurate values for the vibrational energies, the PEC/PES needs to be augmented with corrections accounting for adiabatic, nonadiabatic, relativistic, and QED effects. Corrections due to the finite sizes of the nuclei may also be included. Such an augmented PEC/PES is then used as the potential in the Schrödinger equation, which describes the motion of the nuclei. The solutions of this equation are rovibrational states of the molecule. The states corresponding to the zero rotational quantum number are the pure vibrational states, which we obtain directly in our non-BO calculations. The comparison of our non-BO energies of the pure vibrational states and the energies of those states obtained in the conventional approach, which involves the PEC/PES, can be done at two levels, "nonrelativistic” and "relativistic”. The nonrelativistic-level conventional calculation involves generating a PEC/PES, which includes nonrelativistic electronic energies and adiabatic and nonadiabatic corrections, and using this PEC/PES in the nuclear Schrödinger equation. As noted, the nonrelativistic energies in the non-BO calculations are obtained directly by solving the all-particle Schrödinger equation with the internal Hamiltonian. The conventional approach and our non-BO approach should yield almost identical energy values (see the examples, which are discussed next). The difference is only due to higher order non-adiabatic corrections, which in the conventional approach are accounted for to either second or third order, but in our variational non-BO approach to infinite order. The relativistic-level calculation, which uses the conventional approach, involves an additional perturbation calculation of the relativistic corrections to the PEC/PES electronic energies and using the PEC/PES which includes these corrections in the nuclear Schrödinger equation to determine the rovibrational levels. In our non-BO approach the relativistic corrections to the energies of the vibrational states are obtained as expectation values of the relativistic operators calculated using



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the all-particle wave functions obtained in the non-BO calculations. Thus, the results obtained using the conventional approach and our non-BO approach are not strictly identical. The difference may appear due to two reasons. The first one is due to the non-BO results including relativistic contributions due to both nuclei and electrons, while the conventional results include only the electronic relativistic corrections. The second reason is due to the vibrational wave functions in the conventional approach being generated in the potential which includes the electronic relativistic corrections, while generating the wave function in the non-BO approach (a part of this wave function describes the vibrational state of the molecule) does not include any relativistic effects. These effects are subsequently added to the energies as first-order perturbations. In order to make our non-BO approach more similar to the conventional approach, one would need to include in the internal Hamiltonian used in the non-BO calculations operators representing the relativistic interactions (in a scalar form). Such an approach may be considered in our future work. The molecular non-BO calculations are usually more time consuming than the calculations for m

atoms. The presence of high powers of r1 in the r1 k factor in the diatomic basis functions results in longer calculations of the Hamiltonian matrix elements and the energy gradient. Also, the range of the r1 powers usually increases when the level of the vibrational excitation increases because the r1mk factor has to describe a larger number of radial nodes in the wave function. Thus calculations of higher excited vibrational states require more basis functions than the lower-lying states to achieve the same accuracy level. As mentioned, the non-BO nonrelativistic calculations are done using the standard variational method. Each state is calculated separately from other states. In the calculation the ECG basis set for that particular state is generated and optimized. The optimization involves the ECG exponential parameters, the powers of r1 in the preexponential factor, and the linear coefficients in the expansion of the wave function in terms of ECGs. The exponential-parameter optimization is performed with the aid of the analytic energy gradient determined with respect to these parameters.


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H2 , HD, D2 , T2 , and HeH+ The first set of results we find interesting to compare concerns the dissociation energies of twoelectron systems, H2 , HD, D2 , T2 , and HeH+ . These quantities were calculated using our non-BO approach 12,20,22–25 and by others using the conventional BO approach. 17–19,21,26 The comparison of the results are shown in Table 1.



Table 1: Comparison of the dissociation energies of H2 , HD, D2 , T2 , and HeH+ , and individual contributions to these energies from the non-relativistic energies (Dnr ) and relativistic corrections (total rel. corr. = MV + (d1+d2) + SS + OO; where d1 and d2 are one- and two-particle Darwin corrections). All quantities are in cm−1 . H2 Dnr MV d1+d2+SS OO total rel. corr. D rel

present work 36118.79774 4.4246 -5.5378 0.5442 -0.5691 36118.2287

HD 17,18

19

12,20

36118.7951 4.4583 -5.5209 0.5295 -0.5330

36118.7977 4.4273 -5.5014 0.5422 -0.5319 36118.2658

36406.5105 4.4520 -5.5287 0.5456 -0.5311 36405.9794

17,18

4.4894 -5.5551 0.5347 -0.5310

D2 21

22

36406.5108

36749.0910 4.5102 -5.5865 0.5473 -0.5290 36748.5620

-0.5299 36405.9809

18

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HeH+

T2

17,18

19

22

4.5277 -5.5965 0.5401 -0.5287

36749.091 4.5125 -5.5866 0.5465 -0.5276 36748.5634

37029.2249 4.5479 -5.6241 0.5488 -0.5274 37028.6974

17,18

4.5599 -5.6310 0.5440 -0.5271

23–25

26

14874.653 -1.504

14874.677

0.011 -0.503 14874.150

-0.459 14874.218

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The following conclusions can be drawn from the data presented in Table 1: • In general, the relativistic corrections obtained using the conventional approach are close to the values obtained with the non-BO approach. However, as expected, the differences are larger for the lighter system, H2 , and smaller for the heavier systems, D2 , T2 , etc. The total non-BO relativistic correction to the dissociation energy of H2 differs from the result of Wolniewicz 17,18 by 0.036 cm−1 and from the result of Piszczatowski et al. 19 by 0.0372 cm−1 . This is an expected effect resulting from the non-BO results being dependent on the nuclear masses and the results obtained with the conventional method being independent on the masses in the first order, as they are obtained within the BO-approximation framework. When one of the nuclei in H2 is replaced by a deuteron the total relativistic correction calculated using the non-BO approach of -0.5311 cm−1 becomes very similar to the correction obtained with the BO approach of -0.5310 cm−1 17,18 and -0.5299 cm−1 . 21 This trend continues for D2 where the total relativistic non-BO correction is equal to -0.5290 cm−1 22 and the correction obtained with the BO approach is equal to -0.5287 cm−1 17,18 and -0.52747 cm−1 . 19 A similar small difference occurs for T2 (see Table 1). • Regarding the HeH+ results, an important remark needs to be made. For this system the dissociation energy calculate based on our results is equal to 14874.150 cm−1 and not 14873.8360 cm−1 as mentioned by Pachucki and Komasa. 26 The relatively large difference between these two values of -0.82 cm−1 , which is largely due to the difference between their relativistic correction and what they took as our relativistic correction, was interpreted by Pachucki and Komasa as an indication of a problem in our calculation of the relativistic correction. However, as shown in Table 1, our total relativistic correction of -0.503 cm−1 is different from their correction of -0.459 cm−1 26 by only 0.044 cm−1 . This difference has the same magnitude as the difference in the non-relativistic contributions obtained with the two methods which is equal to 0.024 cm−1 . It is possible that the difference in the basis functions used in the two calculations, which may have different sensitivity to the singularities of the operators, particularly those in the Darwin correction, contributes to the difference in the 19 ACS Paragon Plus Environment


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convergence of the relativistic correction. • The comparison presented in Table 1 indicates that the results obtained with the two approachs, the one involving the BO approximation and the one without it, produce consistent set of results. The agreement is better for heavier systems than for the lightest one, the H2 molecule. The ultimate verification of which approach is more accurate can only come from a comparison with the experimental results. This is however not possible at present because the algorithms for calculating the QED corrections are not yet implemented within our non-BO scheme.

HD+ The the HD+ ion is a good model system to demonstrate the similarities and the differences between the the energies of pure vibrational states obtained with the conventional approach and with our non-BO approach. A peculiarity of HD+ is that in the lowest vibrational states the system is covalently bounded while in the top two states it is ionic. This is related to the asymmetry of the distribution of the electron density among the two nuclei caused by slightly lower energy the electron has when it approaches the deuteron than when it approaches the proton (the total energy of D is slightly lower than the energy of H). As a result the lowest-energy dissociation channel of HD+ is D+H+ and not H+D+ . Our calculations 9 showed that the asymmetry progressively increases as the vibrational excitation of the ion increases and in the top two states the electron is much closer to deuteron than to the proton. In Table 2 (taken from Ref. 9 ) we show the average interparticle distances in HD+ for some selected lowest and highes pure vibrational states calculated as expectation values of the distances over the non-BO wave functions obtained for those states. The charge asymmetry can be seen by comparing the average proton-electron (p − e) distance with the deuteron-electron (d − e) distance. For the highest v = 22 state these distances are 28.46 and 1.600 a.u., respectively, indicating that the system can be described as a D atom interacting with a distant proton. It should be noted that the HD+ charge asymmetry is a purely non-adiabatic effect. Such an effect can be only accounted for in the conventional approach when the non-adiabatic 20 ACS Paragon Plus Environment

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correction to electronic wave function is calculated. Table 2: Expectation values of the deuteron–proton distance, rd−p , the deuteron–electron distance, rd−e , and the proton–electron distance, r p−e , and their squares for the vibrational levels of HD+ at the rotational ground state. All quantities in a.u.

v 0 1 2 3 4 ... 19 20 21 22 D atom ∗ ∗

rd−p 2.055 2.171 2.292 2.417 2.547

hrd−e i 1.688 1.750 1.813 1.880 1.948

r p−e 1.688 1.750 1.814 1.881 1.950

7.099 8.550 12.95 28.53

4.198 4.569 2.306 1.600 1.500

4.421 5.516 12.19 28.46

In the ground state

Questions which now arise are how the charge asymmetry can be accounted for in the calculation of the relativistic corrections and how this effect affects the values of the corrections. To answer these questions we have performed calculations of all leading relativistic corrections for all 23 vibrational states of HD+ and the results are shown in Table 3.



Table 3: Comparison of the total non-relativistic (nr) non-adiabatic (non-BO) variational energies of all pure vibrational states of HD+ obtained in the present work with the energies obtained by Moss 28 and comparison of the total relativistic corrections (rel cor) and individual contributions to these corrections (mass velocity (mv), one- and two-particle Darwin (d1+d2), and orbit-orbit (oo)) with the values obtained by Wolniewicz and the values obtained by Howells and Kennedy. The total energies and the relativistic corrections are in cm−1 . Darwin corrections are determined without the magnetic moment anomaly. The v = 0 and 1 states were calculated with 1000 ECGs, the v = 2, . . . , 21 states with 2000 ECGs, and the v = 22 state with 4000 ECGs.

22

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v 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

D + H+ D+ + H ∞ H + H+

nr energy present work -131223.4362 -129310.4648 -127493.6254 -125770.0574 -124137.2749 -122593.1563 -121135.9388 -119764.2148 -118476.9358 -117273.4185 -116153.3562 -115116.8377 -114164.3686 -113296.9086 -112515.9033 -111823.3402 -111221.8057 -110714.5587 -110305.5753 -109999.5291 -109801.4904 -109717.6363 -109707.8554 -109707.4265

Moss 28 -131223.4363 -129310.4649 -127493.6255 -125770.0575 -124137.2750 -122593.1566 -121135.939 -119764.2152 -118476.9364 -117273.4192 -116153.3571 -115116.8388 -114164.371 -113296.9117 -112515.9068 -111823.3456 -111221.8153 -110714.5702 -110305.5869 -109999.5438 -109801.502 -109717.6423 -109707.8575

mv -9.1988 -8.9886 -8.8012 -8.6224 -8.4547 -8.2995 -8.1592 -8.0275 -7.9069 -7.7995 -7.6988 -7.6118 -7.5258 -7.4618 -7.4005 -7.3498 -7.3079 -7.2749 -7.2556 -7.2484 -7.2589 -7.2773 -7.2947 -7.2966 -7.2887 -7.3046

present work d1+d2 7.6015 7.4244 7.2658 7.1131 6.9691 6.8356 6.7138 6.5984 6.4919 6.3962 6.3050 6.2251 6.1436 6.0825 6.0213 5.9681 5.9213 5.8810 5.8513 5.8309 5.8259 5.8248 5.8375 5.8389 5.8341 5.8437

oo -0.0056 -0.0055 -0.0053 -0.0052 -0.0051 -0.0050 -0.0049 -0.0049 -0.0048 -0.0047 -0.0047 -0.0046 -0.0046 -0.0046 -0.0046 -0.0046 -0.0046 -0.0046 -0.0046 -0.0046 -0.0045 -0.0035 -0.0032 -0.0032 -0.0064 0

rel cor -1.6029 -1.5697 -1.5407 -1.5145 -1.4907 -1.4689 -1.4504 -1.4340 -1.4198 -1.4080 -1.3986 -1.3914 -1.3869 -1.3839 -1.3838 -1.3863 -1.3911 -1.3984 -1.4088 -1.4221 -1.4375 -1.4559 -1.4604 -1.4609 -1.4609 -1.4609

mv -9.2190 -9.0123 -8.8199 -8.6410 -8.4750 -8.3213 -8.1793 -8.0486 -7.9288 -7.8196 -7.7207 -7.6320 -7.5532 -7.4844 -7.4253 -7.3762 -7.3371 -7.3081 -7.2895 -7.2816 -7.2844 -7.2961


Wolniewicz 17,27 d1+d2 7.6169 7.4422 7.2789 7.1264 6.9842 6.8518 6.7287 6.6144 6.5088 6.4114 6.3220 6.2405 6.1666 6.1002 6.0413 5.9898 5.9458 5.9093 5.8804 5.8595 5.8467 5.8424

rel cor -1.6022 -1.5702 -1.5411 -1.5147 -1.4908 -1.4695 -1.4507 -1.4342 -1.4201 -1.4083 -1.3988 -1.3916 -1.3867 -1.3842 -1.3841 -1.3865 -1.3914 -1.3989 -1.4092 -1.4222 -1.4378 -1.4537 -1.4605

Howells and Kennedy 29 rel cor -1.6015 -1.5696 -1.5405 -1.5141 -1.4904 -1.4691 -1.4503 -1.4340 -1.4199 -1.4081 -1.3986 -1.3915 -1.3867 -1.3842 -1.3841 -1.3864 -1.3913 -1.3989 -1.4091 -1.4221 -1.4377 -1.4564

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The following conclusions can be drown by examining the results presented in Table 3: • The total non-relativistic energies obtained with our non-BO approach agree with the energies obtained with Moss 28 using the conventional approach. The agreement is much better for the lowest states than for the higher ones. This shows that more ECGs need to be used in the non-BO calculations of the higher states. • The values of the total relativistic correction obtained in the present work compares very well with the results obtained by Wolniewicz 17,27 and Howells and Kennedy. 29 For all states the difference is only in the fourth digit. • The individual relativistic corrections for the v = 22 state obtained in our calculations (mv = −7.2947 cm−1 , d1 + d2 = 5.8375 cm−1 , and oo = −0.0032 cm−1 ) are, as expected, much closer to the values obtained for D+H+ (mv = −7.2966 cm−1 , d1 + d2 = 5.8389 cm−1 , and oo = −0.0032 cm−1 ) than to the values for D+ +H (mv = −7.2887 cm−1 , d1 + d2 = 5.8341 cm−1 , and oo = −0.0064 cm−1 ). However, interestingly, even though the individual contributions to the relativistic correction are different for D+H+ and D+ +H, their sums for the two systems are almost the same. They are also almost the same as the value obtained for ∞ H+H+ , i.e. for a system with infinite nuclear masses. • In the calculations by Wolniewicz the v = 22 state is missing. Also, his value of the relativistic correction for the v = 21 state of 1.4537 cm−1 is noticeably different from the values obtained in our calculation and in the calculation of Howell and Kennedy (1.4559 and 1.4564 cm−1 , respectively). This indicates that, perhaps, the PEC used by Wolniewicz was somewhat inaccurate for larger values of the internuclear distance.

Summary The leading relativistic corrections for molecular systems can be calculated using various approaches. They include Dirac-Fock calculations followed by relativistic CI (RCI), relativistic 23 ACS Paragon Plus Environment


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CASSCF and CASPT2, often with DKH transformation, and relativistic density functional theory. In very accurate calculations of potential energy surfaces for small molecular systems the relativistic corrections are usually calculated using the first-order perturbation theory as expectation values of the operators representing the corrections with the nonrelativistic wave function of the system. In the conventional approach based on the Born-Oppenheimer (BO) approximation, the calculation is done using the nonrelativistic electronic wave function. When the BO approximation is not assumed and all particles forming the molecular systems are threated on equal footing, the wave function obtained for the system depends on the relative coordinates of all particles comprising the system. Therefore the lowest-order relativistic corrections are determined as expectation values involving the wave function which describes the state of both electrons and nuclei. We have developed a method for such calculations where the non-BO wave function are expanded in terms of explicitly correlated Gaussian functions. In the first numerical example shown in this work we compare the relativistic corrections to the dissociation energies of the H2 , HD, D2 , T2 , and HeH+ two-electron systems obtained with the BO and non-BO approachs. It is shown that, as expected, the deviation between the two sets of results increases with decreasing masses of the nuclei. For the heavier isotopologues of H2 , the deviations are very small, but become more noticeable for the lightest isotopologue, H2 . Next, the relatistic corrections are calculated for the HD+ ion It is shown that our non-BO approach produces relativistic corrections, which are very similar to those obtained by others using the conventional approach based on the BO approximation. It it also shown that the individual contributions to the relativistic correction for the top vibrational states calculated using the non-BO approach converge to values of the contributions obtained for the D atom reflecting the asymmetry of the charge associated with the fact that the lowest-energy dissociation channel for the HD+ ion is D + H+ . If those contributions are calculated using the conventional BO approach, they converge to values being averages of the D and H values. To make the calculations of ground- and excited-state energies of molecules more exact in the framework of the non-BO approach leading QED corrections need to be calculated. We will do


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that in the near future using the approach involving the spectral representation method described in our recent work. 30



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data

was

derived

based

on

the

information

provided

in:

http://www.fizyka.umk.pl/ftp/pub/publications/ifiz/luwo/. (18) Wolniewicz, L. Nonadiabatic Energies of the Ground-State of the Hydrogen Molecule, J. Chem. Phys. 1995, 103, 5, 1792-1799. (19) Piszczatowski, K.; Lach, G.; Przybytek, M.; Komasa, J.; Pachucki, K.; Jeziorski, B. Theoretical Determination of the Dissociation Energy of Molecular Hydrogen, J. Chem. Theory and Comput. 2009, 5, 11, 3039-3048.



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Phys. 1991, 74, 1, 103-111. (28) Moss, R.E. Calculations for Vibration-Rotation Levels of HD+ , in Particular for High n Mol. Phys. 1993, 78, 2, 371-405. (29) Howells, M.H.; Kennedy, R.A. Relativistic Corrections for the Ground and 1st Excited-States + + of H+ 2 , HD , and D2 , J. Chem. Soc. Faraday Trans. 1990, 86, 21, 3495-3503.


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(30) Stanke, M.; Adamowicz,L.; Ke¸dziera,D. Selection of a Gaussian Basis Set for Calculating the Bethe Logarithm for the Ground State of the Hydrogen Atom, Mol. Phys. 2013, 111, 1-6.



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Molecular Relativistic Corrections Determined in the Framework

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