1 Introduction

A thermodynamic process is the evolution of the thermodynamic state of a system. In classical thermodynamics, the state of a system is described by thermodynamic quantities such as the temperature T, the entropy S, the pressure p and the volume V. Keeping one of these quantities constant makes up a basic classical thermodynamic process, e.g. constant T or V results in an isothermal or isochoric process, respectively. Characterized by discrete levels, correlations, size effect, etc., quantum systems are expected to show a thermodynamic behavior which considerably differs from the description of classical thermodynamics [1–4]. Some of the thermodynamic quantities can be statistically generalized to typical quantum model systems described by a set of energy levels {E i } and occupation probabilities {P i } [5], which opens the possibility of building quantum thermodynamic processes in analogy to the classical ones, where certain quantities are kept constant [6–12].

Characterized by a constant pressure the isobaric process is a crucial building block of both the Diesel and the Brayton cycles. With respect to the energy transformation during an isobaric process work and heat exchange occur at the same time. Therefore, the realization of a quantum isobaric process involves the generalization of the notion of pressure within a quantum context. In one-dimensional systems, the pressure p and the volume V can be replaced by the force F and length L. Using {E i } and {P i } one can write the force as \(F=- \sum{_{i}}P_{i}\frac{dE{_{i}}}{dL}\). Since the process proceeds through adjusting L, \(\frac{dE{_{i}}}{dL}\) generally change and a constant force requires {P i } to vary accordingly. Isobaric processes were already proposed for model systems such as the one-dimensional potential well with a single-mode radiation field, where {P i } were tuned through a thermal bath [10].

In the present work, we extend this idea to a realistic system by numerically building isobaric processes on a Ni2 dimer. Real molecules have more complicated electronic structure and more interesting properties than model systems, a fact which can also lead to various applications. As a first study, implementing isobaric processes in Ni2 is expected to give insight into the thermodynamic properties of quantum magnetic systems.

2 Level Scheme of Ni2

The high-level symmetry-adapted-cluster configuration interaction (SAC-CI) method [13] is used to calculate the excited electronic states of Ni2. Our calculations are performed with the GAUSSIAN09 package and the Los Alamos basis set [14]. The energies of the excited states of Ni2 are calculated while varying the interatomic distance from 1.9 to 3.4 Å in steps of 0.05 Å. 22 singlet and 22 triplets excited states are calculated at the SAC-CI level. Figure 1(a) shows the level energies as functions of the bond length for the lowest 17 states, 9 of which are triplets. The ground state is a B1u triplet state and reaches its minimal energy at 2.56 Å, which is in reasonable agreement with experimental data [15]. When varying the interatomic distance a considerable amount of level-crossing takes place. The energy–bond length (E–R) curves are interpolated using cubic splines and the force of each state is calculated as \(F_{i}=-\frac{dE{_{i}}}{dR}\).

Fig. 1
figure 1

The total energies of the many-body electronic states of Ni2 with respect to bond length. (a) SAC-CI calculated energies. (b): Effect of SOC and magnetic field included

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Based on the SAC-CI wavefunctions, spin-orbit coupling (SOC) and a static external magnetic field are introduced perturbatively [16]:

(1)

This Hamiltonian is diagonalized in the basis of the many-body wavefunctions obtained from the SAC-CI calculations. As shown in Fig. 1(b), the degeneracy of the SAC-CI triplet states is lifted giving rise to 35 states, which are used to introduce the isobaric processes in the following sections.

3 Isobaric Process

3.1 Boltzmann Scheme

The occupation probabilities {P i } of each excited state of Ni2 are thermalized. For each bond length R, which corresponds to a certain level scheme {E i (R)}, {P i } are determined from a thermalization temperature T and a Boltzmann distribution

$$P_i(R,T)={e^{-\frac{E_i(R)}{kT}}}/ {Z(R)} , \quad Z(R)= \varSigma{_i} {e^{-\frac{E_i(R)}{kT}}} $$

where k is the Boltzmann constant. Given the starting geometry R 0 and the initial temperature T 0, the force can be calculated as:

$$F_0=F(R_0)=-\sum_i P_i(R_0,T_0)\frac{dE{_i}(R)}{d R} \bigg\vert_{R_0}. $$

As R varies in the process, T needs to be changed in order to keep the force at the constant value of F 0:

This actually renders the temperature T a function of F 0 and R: T=T(E(R),F 0), with initial value T 0=T(R 0,F 0). To numerically determine this function, the whole range of R is divided into small intervals of 0.002 Å and the non-linear equation F(R,T)−F 0=0 is solved at each point. This is done numerically with the secant method since the dependence of the energies E i (R) on the bond length R cannot be expressed in an analytic form.

3.2 Isobaric Range

From the above discussion one sees that, given the initial temperature T 0 and the starting bond length R 0, the temperature in an isobaric process is actually determined by the levels {E i (R)} and their derivatives {F i (R)} for each bond length R. The total force F, being a convex sum, cannot exceed the maximal value of its summands: F(R)≤max{F i }. The Boltzmann distribution imposes an additional restriction to the value of F(R). In realistic systems, the geometry dependence of energy E i (R) can be so complicated that the non-linear equation F(R,T)=F 0 has no solution. As a result, starting at R 0 the isobaric process of the dimer can proceed only until the bond length reaches a limiting value, beyond which the force cannot be kept constant any longer. Usually, we find two such limiting values on both sides of R 0, which actually define a continuous range covering R 0. We name this the isobaric range, in which the isobaric process starting from R 0 with temperature T 0 is possible. This restriction on the isobaric processes actually results from the variation of the energy derivatives of the excited states with the bond length, which, in turn, originates from the electronic structure of Ni2.

Another limiting factor of the isobaric process is the temperature. For the excited states considered in this work the energy difference between the ground and highest electronic state involved is typically of the order of 1.0 eV. To keep most of the population in these states it is necessary to put a limit on the value of the temperature T, which, in turn, confines the variation of the bond length in a range, since for a certain force F 0 the temperature T virtually depends on the bond length R. In our work, the temperature is kept below 4600 K in order to ensure that at least 99 % of the electronic population is held in the states considered. With this upper limit, the bond length R for the isobaric process is determined as follows:

For an isobaric process starting with the given temperature T 0 and bond length R 0 at the ith node, the temperatures are calculated on the nodes of both sides using the algorithm mentioned above. The calculations are first performed for the nodes next to the ith node, namely the (i+1)th and (i−1)th nodes. If the value of the temperature can be found and is below 4600 K, it is within the isobaric range of the initial bond length. Then the calculation is carried on to the (i+2)th or (i−2)th node until the temperature exceeds the upper limit or the equation has no solution. This way, we acquire the temperature values for each node and the isobaric range which covers the starting bond length.

3.3 Heat and Work

The increments of work and heat in a quantum context are:

The work is interpreted as the energy change caused by the shift of the energy levels while the distribution profile remains the same, and the heat exchange is caused by the change of distribution with the level scheme unchanged.

In an isobaric process, each infinitesimal variation of R shifts the energy levels and the constant force requires {P i } to change accordingly, which explains why the work and the heat in an isobaric processes cannot be considered separately. According to the quantum interpretation, the work and the heat are calculated for each small change of the bond length. This means that for the step between jth and (j+1)th node,

where \(P_{i}^{(j)}\), \(E_{i}^{(j)}\) are the population and the energy of ith level at the jth node, respectively. In this formalism, the heat absorbed (Q in) and the work performed to the dimer (W in) have positive values, while the heat given off (Q out) and the work performed by the system (W out) are negative.

3.4 Isobaric Processes in Ni2

One of the fundamental differences between the quantum system Ni2 and macroscopic systems (such as the ideal gas) is that the force on the Ni2 dimer can change its sign for bond lengths smaller and larger than the equilibrium value. In our system all forces F i have different signs for R smaller than 2.20 Å and larger than 2.65 Å because the energetic minima of all states lie within this range. This fact has two effects on the isobaric process: (i) the isobaric range is generally outside the range 2.20–2.65 Å (either side) and can only slightly penetrate it from one side, and (ii) there exists the possibility of realizing isobaric processes with different directions of force using one single system. In order to observe the isobaric processes with both negative and positive forces we choose two starting positions (R 0=2.05 and 2.65 Å), one on each side of the range 2.20–2.65 Å.

Figure 2 shows the variations of the temperatures for isobaric processes starting at 2.05 Å and with different initial temperatures T 0. For each process the isobaric range is also marked. For the isobaric process starting at 2.05 Å and under 800 K, the isobaric range is 2.042–2.096 Å. The temperature at the left end is 481.59 K and at the right end 290.60 K. The peak value of 1066.03 K occurs at 2.068 Å. The initial temperature of 1000 K extends the isobaric range from 2.036 to 2.100 Å and the highest temperature is 1452.30 K at 2.084 Å. The initial temperature of 1200 K further extends the isobaric range to 2.034–2.116 Å. In this range, the temperature needed to maintain the force constant monotonically increases up to 4503 K.

Fig. 2
figure 2

Variation of temperatures in the isobaric processes starting at R 0=2.05 Å with initial temperature T 0=800,1000,1200 K. The starting position is marked by R 0 and the isobaric range by the dashed lines at the ends of the temperature curves. See also text

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Due to its strong level-dependence the variation of the temperature T(R)=T(E(R),F 0) is determined by the evolution of the energy (or energy derivative) on the isobaric range. Given the ranges above it also depends on the initial temperature T 0. At the starting point R=2.05 Å the states higher in energy contribute more to the force due to their steeper slopes. A high initial temperature leads to increased occupations of the higher levels and a larger total force. As the bond length is extended, the derivatives of the energies decrease. To compensate this reduction and maintain the force constant, the energetically higher states need to be weighted heavier in the total sum, which results in increased temperatures. For relatively low initial temperatures, the occupation of the high-energy levels is less important and the temperature largely depends on the variation of the derivatives of the low-lying states.

Figure 3 shows the work and the heat for the processes in Fig. 2 as functions of the bond length from R 0 to the large ends. The linear behavior of the work is normal for the isobaric processes and the steepest slope for T 0=1200 K indicates the largest force. The heat absorption, as can be seen, is strongly influenced by the behavior of the temperature function. Given the Boltzmann distribution used in this work the heat, or the change of the population among the states, can be caused by both shift of energy levels and change in temperature. In the short isobaric range of less than 0.1 Å, the level shift is typically around 0.2 eV and can hardly affect the distribution profile, which renders the temperature the dominant factor.

Fig. 3
figure 3

The variation of the work and the heat in the isobaric processes shown in Fig. 2 with R extending from R 0 to larger ends

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For the other starting position R=2.65 Å, the initial temperatures are set to be 1600, 1800, and 2000 K (see Fig. 4). The first two temperatures lead to similar isobaric ranges and behavior of the temperature function. Both isobaric ranges expand from slightly smaller than 2.65 Å to around 2.72 Å, where the temperatures needed to maintain a constant force increase first moderately and then abruptly up to the 4600 K limit. In comparison, the isobaric range for T 0=2000 K not only roughly covers that range but expands to 2.60 Å on the other side with the temperature rising up to 4500 K. The work and heat are also calculated for the bond-extending isobaric processes and are shown in Fig. 5. The positive values indicate that work is actually performed to the system and increases the total energy. However, the energy gained through heat absorption is nearly one-order of magnitude larger than the energy gained from work, which results from the high temperatures involved.

Fig. 4
figure 4

Same as Fig. 2 but with R 0=2.65 Å and T 0=1600,1800,2000 K

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Fig. 5
figure 5

Same as Fig. 3 but with R 0=2.65 Å and T 0=1600,1800,2000 K

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4 Spin Evolution

Incorporating spin–orbit coupling and a static magnetic field the spin moment for each of the 35 states {S i } is calculated at each bond length. The total spin moment of Ni2 is S=∑ i P i S i . The dramatic modification of the distribution profile during the isobaric processes also leads to a change of the spin moment of the Ni2, which actually embodies an observation of the thermodynamic behavior of spin.

However, for the isobaric processes starting at 2.05 Å, the spin evolution can hardly be expected to be meaningful. As shown in Fig. 1(a), the lowest triplet state (3B1u) crosses most of the other states in the range between 1.90 and 2.05 Å. As a result, in this range the perturbative inclusion of spin–orbit coupling and static field mixes this state with different ones, thus each of the resulting states may change its spin moment at different bond lengths. On the other hand, for the isobaric processes starting at 2.65 Å, the spin moment remains constant. Figure 6 shows the spin moments S of all 35 states for bond lengths R=2.65, 2.70 and 2.75 Å. As can be seen, the three spin moment profiles are similar except for a few states, for which the spin moment values of R=2.70 Å lie between those of R=2.65 Å and R=2.75 Å. For the qualitative observation of the variation of the total spin within the range 2.65–2.75 Å one can safely use the spin profile at R=2.70 Å.

Fig. 6
figure 6

The spin moment of the 35 states at the bond lengths R=2.65,2.70,2.75 Å. The dashed lines are guides to the eye

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Figure 7 shows the spin moments in the isobaric processes against bond length from 2.65 to around 2.72 Å. At R=2.65 Å the initial spin moment is lower for higher temperatures. For all three initial temperatures, the magnitude of the spin moments decreases to less than half. Understanding the spin behavior is straightforward. At low temperatures the ground state, the spin moment of which is nearly −1, contributes the most to the total spin moment. During the isobaric processes the increasing temperature tends to average out the distributions among the states which, in turn, decreases the net spin.

Fig. 7
figure 7

Variation of the total spin moment of Ni2 during the isobaric process starting at R 0=2.65 Å and extending to larger ends

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5 Summary

In this work, we numerically realize the isobaric processes in the Ni2 dimer. With the pressure replaced by the force and the volume by the bond length in this one-dimensional system, the isobaric processes are realized by exploiting the population on the excited states of the Ni2, i.e. by extending or compressing the bond length of the dimer while tuning the temperature. The Ni2 dimer has a more complicated electronic structure than a simple quantum model and a more complex behavior than its classical, macroscopic counterparts. Therefore, given some initial values of bond length and temperature, the isobaric process can take place only within a certain range. This range can only be on one side of the equilibrium length at a time. At a certain bond length, the temperature needed to keep the force constant strongly depends on the local energy derivative, so a thermally-driven isobaric process needs the detailed knowledge of the energy evolution across the whole isobaric range. The temperature variation, the heat, and the work are discussed for two sets of isobaric processes with different signs of the force. Finally, it is found that during the isobaric processes with expanding bond length the total spin moment of the Ni2 is reduced with increasing temperature.