^{†}Corresponding author. Email: qhx@whu.edu.cn
^{*}Project supported by the National Natural Science Foundation of China (Grant No. 11274246.)
Based on the Bspline basis method, the properties of the helium atom confined inside an endohedral environment, such as buckminster fullerene, are studied. In our calculations, the endohedral environment is a parabolic potential well. In this situation, the phenomenon of “mirror collapse” is exhibited for energy levels of a confined helium atom. The “giant resonance” of oscillator strength of the dipole transition emerges with the variation of depth of the confining well. The physical mechanisms of these phenomena are analyzed in this paper.
With the success of La@ C_{60}^{[1]} preparation, the experimental research has successively obtained K@ C_{60}, ^{[2]} He@ C_{60}, ^{[3]} and Ne@ C_{60}, ^{[4]} and so on. In such an environment, which is a doublewell of the atomic potential, some novel properties of atoms appear. In the theoretical research of endohedrally confined atoms, ^{[5– 17]} only Connerade, ^{[7, 9, 11, 13]} Qiao, ^{[14– 16]} and Zhang^{[17]} have focused on their spectral properties. Meanwhile, some researchers have done some experiments on the spectral properties of endohedrally confined atoms.^{[18– 23]} In our work, a Bspline functions^{[24]} method is applied to calculate the energy level and the dipole oscillator strength of endohedrally confined helium, when the square potential well is changed into a parabolic potential one. The causes of the “ mirror collapse” and “ giant resonance” phenomena are analyzed.
This paper is organized as follows. In Section 2, the confined helium model and the Bspline functions are introduced. The results of energy levels and oscillator strength of the endohedrally confined helium are given in Section 3. This paper is closed by the conclusions in Section 4.
For a helium atom in an endohedral environment, the Hamiltonian is:
where V(r) is the attractive potential shell. In this paper, the shape of the attractive potential is changed to soften the endohedral environment. The attractive potential shell is defined as a parabola form
where the parameters r_{A} and r_{B} are the inner and outer radii of the confining potential well, and the values are 5.75 a.u. (the unit a.u. is the abbreviation of atomic unit) and 7.64 a.u., ^{[25]} respectively. In quantum mechanics, the equivalent potential of an electron is
Figure 1 shows the radial distribution of the equivalent potential of the ^{1}P_{1} states. It is obviously a double well structure. The Coulomb well is called the inner well and the confining well is called the outer well. The parameter V_{0} is the depth of the outer well.
Normally, the radial wavefunctions are expanded in three spatial ranges: 0 < r ≤ r_{A}, r_{A} < r ≤ r_{B}, and r_{B} < r ≤ ∞ . In this paper, Bspline functions are used to expand the wavefunctions and the boundary conditions are satisfied very simply.
With the order k, the Bspline functions are defined as
Here, {t_{i}} is the knots sequence set on the x axis. In our work, the order k is set to 7, so the knots sequence is
The basis set is
which is used to expand the wavefunctions of the system. From the Schrö dinger equations, the matrix equation is obtained by
where H and M are respectively the Hamiltonian and overlap matrix. E and C are respectively the eigenvalues and eigenvectors, which can be obtained by solving the matrix equation (11).
In quantum mechanics, the oscillator strength of a dipole transition is calculated by
where Δ E =  E_{f} − E_{i} , J_{i} is the initial state’ s total angular quantum number, D_{if} is the element of the dipole transition matrix and is defined by
As a consequence, for a specific dipole transition, the oscillator strength is determined by two terms: the energy difference and the square of the dipole transition matrix element.
By solving Eqs. (1) and (11), the energy levels of the endohedrally confined helium have been calculated. In our work, 11760 basis sets are included in the expansion of the wave function. As a validation of the Bspline functions methods, we get the ground state energy to − 2.903703 Hartree in the case that the depth of the outer potential well is 0 Hartree. The reliability of the method was verified.
In our calculations, V_{0} is taken from 0 to 6 Hartree in order to study the helium atom confined in buckminster fullerene more generally. The results of the calculation of the energy levels of the ^{1}S_{0} and ^{1}P_{1} states of a confined helium, as a function of the depth of the confining well, are shown in Fig. 2 and Fig. 3.
As shown in Fig. 2 and Fig. 3, the energy levels of the ^{1}S_{0} and ^{1}P_{1} states of helium reveal similar phenomena. Now, as an example, the ^{1}S_{0} states are used to describe the phenomena. It can be seen that a few of ^{1}S_{0} states hold the energies of an unconfined helium atom when V_{0} is small. It seems that the outer confining well has no effect on them. The phenomenon can be understood by the distribution of the radial wavefunctions of these ^{1}S_{0} states. Figure 6(a) shows the onebody densities of the 1^{1}S_{0} and 2^{1}S_{0} states, which can represent the distributions of these states, with V_{0} = 1.0 Hartree. This reveals that the radial wavefunctions of these ^{1}S_{0} states are almost zero in the range of the outer well. So the outer confining well has no effect on them. New ^{1}S_{0} states emerge when the depth of the outer confining well continues to increase, and their radial wavefunctions are distributed in the range of the outer well. Thus, new states are bound in attractive confining wells, but the original states are still bound in Coulomb wells. The new ^{1}S_{0} states will change the existing energy sorting. For example, when a new 1^{1}S_{0} state emerges, the original 1^{1}S_{0} state will become a 2^{1}S_{0} state, and 2^{1}S_{0} states will become 3^{1}S_{0} states, and so on.
In quantum mechanics, the condition that a system has bound states is that the depth of the confining well must achieve a certain value when the width of the confining well is fixed. In the doublewell model, if the Coulomb potential is removed, the bound states will appear only when V_{0} ≥ 1.018 Hartree. To understand the influences of the Coulomb potential and the attractive confined well for the system energies, we use Table 1 and Table 2 to show the relationship between them. In Table 1, the energies of ^{1}S_{0} states of free helium, only confining attractive potential helium (Coulomb potential is removed) and doublewell helium, are given respectively, with V_{0} = 4.0 Hartree. The row of the 1^{1}S_{0} state shows that the energies of the 1^{1}S_{0} state of free helium and doublewell helium are both − 2.9037 Hartree. It seems that the outer well has no effect on the 1^{1}S_{0} state. However, the 2^{1}S_{0} state of doublewell helium is a new state, whose energy is − 2.3380 Hartree. This is the result of the cooperation of the Coulomb potential and the confining attractive potential.
In Table 2, the V_{Coulomb} and V_{onfined} are expectations of Coulomb potential and confining attractive potential with V_{0} = 4.0 Hartree, respectively. R is the ratio of the expectation of the confining attractive potential to the expectation of the total potential. R is calculated by
It can be seen that the Coulomb potential makes the major contribution to the 1^{1}S_{0} state, and the confining attractive potential makes the dominant contribution to the 2^{1}S_{0} state. So the 1^{1}S_{0} state is the original state, and the confining attractive potential contribution can be ignored. The 2^{1}S_{0} state is a new state, and the Coulomb potential contribution is relatively small compared with the confining attractive potential. It is worth noting whether it is a new or an original state, the energy is the result of the combination of Coulomb potential and confining attractive potential.
Now let us focus on the analysis of the ^{1}S_{0} states. The 1^{1}S_{0} state holds − 2.9037 Hartree, which is the 1^{1}S_{0} state energy of an unconfined helium atom. The first ^{1}S_{0} state energy line is a horizontal line, as the depth of the outer attractive well increases until its depth reaches to − 4.773 Hartree. Then, the 1^{1}S_{0} state energy decreases very quickly, and its energy shows linear decreasing with the deepening of the outer confining well. To understand this phenomenon, the onebody density of the 1^{1}S_{0} state is calculated, as a function of the outer attractive well. For V_{0} = − 4.771 Hartree, the onebody density of the 1^{1}S_{0} state is distributed around the nucleus, which indicates the first ^{1}S_{0} state is a free helium ground state, which can be seen from Fig. 6(b). For V_{0} = − 4.773 Hartree, the figure 6(c) shows that the onebody density of the 1^{1}S_{0} state is distributed around the nucleus and the confining well, indicating that the 1^{1}S_{0} state is the compound of a confining state and an unconfined helium ground state. In Fig. 6(d) with V_{0} = − 4.775 Hartree, the onebody density of the 1^{1}S_{0} state is distributed in the range of the outer confining well, indicating that the first ^{1}S_{0} state is a bound state. The second ^{1}S_{0} state has a similar behavior as the first ^{1}S_{0} state has shown formerly. In Fig. 2, its energy holds − 2.1459 Hartree when V_{0} is small, which is the value of the 2^{1}S_{0} state energy of the unconfined helium, and the state stays in the Coulomb well. At the point V_{0} = − 3.728 Hartree, the energy of the second ^{1}S_{0} state suddenly decreases and continues to decrease with the deepening of the outer confining well till V_{0} = − 4.773 Hartree. The state “ collapses” into the confining well. Then, when V_{0} > − 4.773 Hartree, the state “ collapses” into Coulomb potential and stays there. At the point V_{0} = − 4.891 Hartree, the state “ collapses” into the outer attractive well again and stays there finally. Some other ^{1}S_{0} and ^{1}P_{1} states’ behaviors have the same phenomena, which are shown in Fig. 2 and Fig. 3, respectively.
The phenomenon of “ collapse” can also be understood from the ratio of the expectation of the confining attractive potential to the expectation of the total potential. Figure 4 shows that the ratio R is very small for the 1^{1}S_{0} state when V_{0} is small, which implies that the Coulomb potential is dominant for the 1^{1}S_{0} state. At the point V_{0} = − 4.773 Hartree, the ratio R suddenly increases, which indicates that the 1^{1}S_{0} state “ collapses” into the confining well. Then, the ratio R is a large value when V_{0} ≥ − 4.773 Hartree, indicating that the outer attractive well is dominant for the 1^{1}S_{0} state. The 2^{1}S_{0} state’ s ratio reveals the same phenomenon. The ratio R is small when the 2^{1}S_{0} state is bound in the Coulomb well, and it is very large when the 2^{1}S_{0} state is bound in the outer well. At the “ collapse” point, the ratio R suddenly changes. The phenomenon also exists in other states of ^{1}S_{0} and ^{1}P_{1}, which are shown in Fig. 4 and Fig. 5.
At the “ collapse” point, V_{0} = − 4.773 Hartree from Fig. 6(c), the first and the second ^{1}S_{0} states are both strong mixtures of an unconfined helium state and a confined helium state, which leads to the result that their energy lines do not cross. Thus, at the avoided crossing point, V_{0} = − 4.773 Hartree, the first ^{1}S_{0} state suddenly “ collapses” into the outer confining well, which previously is bound in the inner Coulomb well. Simultaneously, the second ^{1}S_{0} state “ collapses” into the inner Coulomb well, which previously is bound in the outer confining well. The behaviors of the ^{1}S_{0} states are called “ mirror collapse.” Figures 2 and 3 show that other states also have the same phenomena. The point of avoided crossing is a cutoff point of the horizontal lines and the oblique lines, which reveals that those states are respectively bound in the inner Coulomb well and the outer confining well. It is worth noting that all oblique lines are parallel, which demonstrates that the energies of all states bound in the outer attractive well decrease at the same rate, i.e. at the same rate as the deepening of the outer well.
The ^{1}S_{0}– ^{1}P_{1} dipole oscillator strengths of the endohedrally confined helium are calculated with Eqs. (12) and (13) in our work. For a certain dipole transition, such as 1^{1}S_{0}– 2^{1}P_{1} dipole transition, the energy difference is certain, so the dipole transition matrix element determines the oscillator strength. The dipole transition matrix element is related to the overlap of the wavefunctions of the states involved in the transition. However, these wave functions are closely related to their onebody densities. So, the dipole transition matrix element is closely related to the overlap of onebody densities of the states involved in the transition.
The 1^{1}S_{0}– 2^{1}P_{1} oscillator strengths are calculated and shown in Fig. 7(a) as functions of the depth of the confining potential well. When V_{0} is a small value, 1^{1}S_{0} and 2^{1}P_{1} states are both bound in Coulomb potential and the overlap of onebody density of them is normal, so that the oscillator strength has the same value corresponding to that of an unconfined helium atom, which can be seen from Fig. 7(a) and Fig. 7(b). At V_{0} = − 4.449 Hartree, the 2^{1}P_{1} state collapses into the outer well, but the 1^{1}S_{0} state is still bound in the Coulomb well. Then the oscillator strength suddenly decreases to a very small value from Fig. 7(a), but it is not zero. The reason is that the overlap of onebody densities of 1^{1}S_{0} and 2^{1}P_{1} states is close to zero from Fig. 7(c) with V_{0} = − 4.6 Hartree. By increasing the depth of the outer well, when the depth V_{0} = − 4.773 Hartree, the 1^{1}S_{0} state collapses into the outer well. In this case, from Fig. 7(d) with V_{0}= − 4.9 Hartree, 1^{1}S_{0} and 2^{1}P_{1} states are all bound in the outer well so that their onebody densities overlap almost completely. So the value of oscillator strength is a large value, which is two orders of magnitude greater than the normal value of an unconfined helium atom. This phenomenon of very strong resonance enhancement of oscillator strength is called “ giant resonance.” At the same time, from Fig. 8(a), the 1^{1}S_{0} and 2^{1}P_{1} states are bound in the Coulomb well when V_{0} is small, so the oscillator strength is normal. At the point V_{0} = − 4.6 Hartree, the 1^{1}S_{0} state is still bound in the Coulomb well, but the 2^{1}P_{1} state is bound in the confining well. At that moment, the oscillator strength is very small. When V_{0} ≥ − 4.773 Hartree, the 1^{1}S_{0} and 2^{1}P_{1} states are both bound in the outer well, the oscillator strength is enhanced and the phenomenon of “ giant resonance” appears. Not only the oscillator strength of 1^{1}S_{0}– 2^{1}P_{1} dipole transition, but also other dipole transitions have the phenomenon about “ giant resonance, ” for example, 1^{1}S_{0}– 3^{1}P_{1} dipole transition, 2^{1}S_{0}– 3^{1}P_{1} dipole transition, and so on. Figure 9 shows the results of the oscillator strength of 1^{1}S_{0}– 3^{1}P_{1} dipole transition and the corresponding onebody densities of 1^{1}S_{0} and 3^{1}P_{1} states in different depths of the outer attractive well. Figure 10 shows the results of the oscillator strengths of the dipole transitions of 2^{1}S_{0}– 2^{1}P_{1} and 2^{1}S_{0}– 3^{1}P_{1}. Figures 8(b), 8(c), and 8(d) show the ratio R of states that are included in the dipole transitions, and corresponding to the 1^{1}S_{0}– 3^{1}P_{1} dipole transition, 2^{1}S_{0}– 3^{1}P_{1} dipole transition, and 2^{1}S_{0}– 3^{1}P_{1} dipole transition, respectively.
The spectral properties of an endohedrally confined helium atom were calculated with the Bspline functions method. With the evolution of the energy spectrum as a function of the depth of the outer attractive well, the phenomenon of “ mirror collapse” exists in endohedrally confined helium. And the phenomenon of “ giant resonance” of the dipole oscillator strength appears with the increase of the depth of the confining well. These phenomena are due to the jumps of the wavefunction between the Coulomb well and the confining attractive well, with the changing depth of the outer attractive well. It is readily apparent that the phenomena can also be understood from the major potential contribution changes between the Coulomb potential and the confining attractive potential, with the deepening of the outer attractive well.
The endohedral environment gives rise to some new questions about confined atoms. For example, the control of laser population, the prominent resonance in the photoionization cross section, and multiple photoionization, with endohedrally confined atoms.^{[26– 28]} Related work is being pursued.
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