# perturbation theory quantum mechanics

An expression for the first-order correction to the pair distribution function of the reference system has also been derived.17, Figure 5.3. (2) into the Schrödinger equation and collecting terms of the same order. 17. Hence, in conventional quantum mechanics, the perturbation theory has, in large, been developed for the systems in which the potentials are real Hermitian that allows only the spectrum of real expectation values for quantum observables. Phase diagram in the concentration-density plane for a binary mixture of non-additive hard spheres with Δ=0.2. New contributor. The ket $$|n^i \rangle$$ is multiplied by $$\lambda^i$$ and is therefore of order $$(H^1/H^o)^i$$. First, we search for the shift of energy as an effect of the perturbation. Reprinted with permission from Abu-Hasanayn, F.; Goldman, A. S.; Krogh-Jespersen, K. Inorg. Equation $$\ref{7.4.13}$$ is the key to finding the first-order change in energy $$E_n^1$$. A critical feature of the technique is a middle step that breaks the problem into "solvable" and "perturbative" parts. The first step in a perturbation theory problem is to identify the reference system with the known eigenstates and energies. Perturbation theory (PT) is nowadays a standard subject of undergraduate courses on quantum mechanics; its emergence is however connected to the classical mechanical problem of planetary motion. While λ introduced in Eq. Estimate the energy of the ground-state wavefunction within first-order perturbation theory of a system with the following potential energy, $V(x)=\begin{cases} The equations thus generated are solved one by one to give progressively more accurate results. One such case is the one-dimensional problem of free particles perturbed by a localized potential of strength $$\lambda$$. It is also the simplest member of a class of ‘core-softened’ potentials that give rise to a rich variety of phase diagrams. However, it is extremely easy to solve this problem using perturbative methods. Such a situation arises in the case of the square-shoulder potential pictured in Figure 5.2. † Cohen-Tannoudji, Diu and Lalo˜e, Quantum Mechanics, vol. Unless otherwise noted, LibreTexts content is licensed by CC BY-NC-SA 3.0. An easy and necessary test of the appropriateness of the PT approximation is simply to investigate important properties (energetic and spectroscopic quantities are preferred over geometric properties, as the latter are often quite insensitive to the computational details) both at the HF and MP2 level of theory. In this method, the potential is split at r = rm into its purely repulsive (r < rm) and purely attractive (r > rm) parts; the former defines the reference system and the latter constitutes the perturbation. After an introduction of the basic postulates and techniques, the book discusses time-independent perturbation theory, angular momentum, identical particles, scattering theory… Consider, for example, the Schrödinger equation initial-value problem. The points are Monte Carlo results and the curves show the predictions of perturbation theory. This limits the range of applicability of the theory at supercritical temperatures.20. Cam-bridge Univ. For that, there are a couple of model problems that we want to work through: (1) Constant Perturbation ψ()t0 = A. which, after combination with the above equation for |ψ〉, yields the formula of interest, This equation is appropriate for an iterative solution. Koga and Morokuma conclude their review by pointing out that for organometallics “…to obtain a reliable energetics, it is necessary to take into account the electron correlation effect, even if the single determinantal wave function is a good starting point.”. (1) is often considered an auxiliary tool that eventually gets substituted as λ=1, it has more than a formal role when studying convergence, vide infra. In yet a third approach the conductance is calculated in a non-perturbation manner between two localized states, rather than between the true bulk states of the tip and sample. Many textbook examples of the utilization of MPn calculations in organometallic chemistry can be found in the classic 1991 review by Koga and Morokuma.18 DFT receives only scant mention in the Koga–Morokuma review. \infty & x< 0 \; and\; x> L \end{cases} \nonumber$. We’re now ready to match the two sides term by term in powers of $$\lambda$$. What is Pk ? Time-independent and time-dependent PT is a classification often used to distinguish the case where stationary solutions are looked for from the situation where Vˆ depends explicitly on time. [6–9] can be suggested in particular, for a meticulous elaboration. Hence, the modeling of dynamical electron correlation and near-degeneracy effects (which is quite common for low coordination number organometallics) requires MC techniques, which are discussed in the following section. Equation $$\ref{7.4.24}$$ is essentially is an expansion of the unknown wavefunction correction as a linear combination of known unperturbed wavefunctions $$\ref{7.4.24.2}$$: \begin{align} | n \rangle &\approx | n^o \rangle + | n^1 \rangle \\[4pt] &\approx | n^o \rangle + \sum _{m \neq n} c_{m,n} |m^o \rangle \label{7.4.24.2} \end{align}, with the expansion coefficients determined by, $c_{m,n} = \dfrac{\langle m^o | H^1| n^o \rangle }{E_n^o - E_m^o} \label{7.4.24.3}$. It should be noted that there are problems that cannot be solved using perturbation theory, even when the perturbation is very weak, although such problems are the exception rather than the rule. The example we choose is that of the Lennard-Jones fluid, a system for which sufficient data are available from computer simulations to allow a complete test to be made of different perturbation schemes.16. {E=\frac{1}{2} h v+\gamma \frac{3}{4 a^2}} share | cite | improve this question | follow | edited Oct 24 at 7:30. user276420. Another point to consider is that many of these matrix elements will equal zero depending on the symmetry of the $$\{| n^o \rangle \}$$ basis and $$H^1$$ (e.g., some $$\langle m^o | H^1| n^o \rangle$$ integrals in Equation $$\ref{7.4.24}$$ could be zero due to the integrand having an odd symmetry; see Example $$\PageIndex{3}$$). Using Equation $$\ref{7.4.17}$$ for the first-order term in the energy of the ground-state, $E_n^1 = \langle n^o | H^1 | n^o \rangle \nonumber$, with the wavefunctions known from the particle in the box problem, $| n^o \rangle = \sqrt{\dfrac{2}{L}} \sin \left ( \dfrac {n \pi}{L} x \right) \nonumber$. Abstract: We discuss a general setup which allows the study of the perturbation theory of an arbitrary, locally harmonic 1D quantum mechanical potential as well as its multi-variable (many-body) generalization. The basic assumption in perturbation theory is that $$H^1$$ is sufficiently small that the leading corrections are the same order of magnitude as $$H^1$$ itself, and the true energies can be better and better approximated by a successive series of corrections, each of order $$H^1/H^o$$ compared with the previous one. Hence, only a small number of terms in the series (12) are needed to calculate the value of y(x) with extremely high precision. New methods are then required, as we discuss in detail in the next section. The curve is calculated from first-order perturbation theory and the points with error bars show the results of Monte Carlo calculations.15, Jean-Pierre Hansen, Ian R. McDonald, in Theory of Simple Liquids (Third Edition), 2006, The λ-expansion described in Section 5.2 is suitable for treating perturbations that vary slowly in space, while the blip-function expansion and related methods of Section 5.3 provide a good description of reference systems for which the potential is rapidly varying but localised. However, changing the sign of $$\lambda$$ to give a repulsive potential there is no bound state, the lowest energy plane wave state stays at energy zero. Nevertheless it is not always justified; here we list some of the reasons why it may break down. In model studies λ occasionally gets in fact tuned to facilitate examination of the PT approximation as a function of perturbation strength. As with Example $$\PageIndex{1}$$, we recognize that unperturbed component of the problem (Equation $$\ref{7.4.2}$$) is the particle in an infinitely high well. Here the following series are assumed. However, this is not the case if second-order perturbation theory were used, which is more accurate (not shown). 20 kJ mol− 1 for Cp models and less (12–16 kJ mol) for larger Cp* derivatives.24 Pyykkö and co-workers have published extensively on aurophilic interactions and have even proposed a recipe for quantification of the aurophilic interaction as the difference between HF and MP2 binding energies.25 Colacio et al.26 have even hypothesized about the utilization of aurophilic attractions, which are thought to be on the order of weak hydrogen bonds, for crystal engineering of Au(i) complexes on the basis of MP2 calculations combined with relativistic pseudopotentials. Switching on an arbitrarily weak attractive potential causes the $$k=0$$ free particle wavefunction to drop below the continuum of plane wave energies and become a localized bound state with binding energy of order $$\lambda^2$$. Note that the zeroth-order term, of course, just gives back the unperturbed Schrödinger Equation (Equation $$\ref{7.4.1}$$). For a broader aspect we refer to the overviews by Killingbeck,22 Kutzelnigg,23 and Killingbeck and Jolicard.24–26, Carl M. Bender, in Encyclopedia of Physical Science and Technology (Third Edition), 2003, Perturbation theory can be used to solve nontrivial differential-equation problems. Thus the sum of the two leading terms is equal to −4.42, whereas the resulted obtained for the total excess free energy from Monte Carlo calculations16 is βF/N = −4.87. Since the pair distribution function has its maximum value in the same range of r, fluctuations in the total perturbation energy WN, and hence the numerical values of F2, are large. where Q(x) is an arbitrary continuous function of x. This is a clear indication that the PT approximation for one or both of the isomers is inappropriate, and one must investigate alternative approaches such as MC techniques. Adding the full expansions for the eigenstate (Equation $$\ref{7.4.5}$$) and energies (Equation $$\ref{7.4.6}$$) into the Schrödinger equation for the perturbation Equation $$\ref{7.4.2}$$ in, $( \hat{H}^o + \lambda \hat{H}^1) | n \rangle = E_n| n \rangle \label{7.4.9}$, $(\hat{H}^o + \lambda \hat{H}^1) \left( \sum _ {i=0}^m \lambda ^i| n^i \rangle \right) = \left( \sum_{i=0}^m \lambda^i E_n^i \right) \left( \sum _ {i=0}^m \lambda ^i| n^i \rangle \right) \label{7.4.10}$. At lower temperatures, however, the results are much less satisfactory. Perturbation theory is a powerful tool for solving a wide variety of problems in applied mathematics, a tool particularly useful in quantum mechanics and chemistry. This effect has been predicted theoretically when the tip–sample separation drops below about 3 Å; it tends to result in a lowering of the potential energy for an electron in the vacuum and a collapse of the tunnelling barrier. FIG. That is, the first order energies (Equation \ref{7.4.13}) are given by, \begin{align} E_n &\approx E_n^o + E_n^1 \\[4pt] &\approx \underbrace{ E_n^o﻿ + \langle n^o | H^1 | n^o \rangle}_{\text{First Order Perturbation}} \label{7.4.17.2} \end{align}, Estimate the energy of the ground-state and first excited-state wavefunction within first-order perturbation theory of a system with the following potential energy, $V(x)=\begin{cases} We begin with a Hamiltonian $$\hat{H}^0$$ having known eigenkets and eigenenergies: \[ \hat{H}^o | n^o \rangle = E_n^o | n^o \rangle \label{7.4.1}$. $E_n^1 = \int_0^L \sqrt{\dfrac{2}{L}} \sin \left ( \dfrac {n \pi}{L} x \right) V_o \sqrt{\dfrac{2}{L}} \sin \left ( \dfrac {n \pi}{L} x \right) dx \nonumber$, or better yet, instead of evaluating this integrals we can simplify the expression, $E_n^1 = \langle n^o | H^1 | n^o \rangle = \langle n^o | V_o | n^o \rangle = V_o \langle n^o | n^o \rangle = V_o \nonumber$, so via Equation $$\ref{7.4.17.2}$$, the energy of each perturbed eigenstate is, \begin{align*} E_n &\approx E_n^o + E_n^1 \\[4pt] &\approx \dfrac{h^2}{8mL^2}n^2 + V_o \end{align*}. † Shankar, Principles of Quantum Mechanics, Ch. Further development of such enhanced DFT approaches to organometallic complexes is of interest. Perturbation theory (in quantum mechanics) is a set of approximation schemes for reducing the mathematical analysis of a complicated quantum system to a simpler mathematical solution. In the separation used by Barker and Henderson13 the reference system is defined by that part of the full potential which is positive (r < σ) and the perturbation consists of the part that is negative (r > σ). We now have two degree-3 internal vertices (labeled by times s and t) and two degree-1 external vertices, both labeled by time 0. Wu, Quantum Mechanics, Ch. The first step in any perturbation problem is to write the Hamiltonian in terms of a unperturbed component that the solutions (both eigenstates and energy) are known and a perturbation component (Equation $$\ref{7.4.2}$$). The work of Barker and Henderson is a landmark in the development of liquid-state theory, since it demonstrated for the first time that thermodynamic perturbation theory is capable of yielding quantitatively reliable results even for states close to the triple point of the system of interest. That is, eigenstates that have energies significantly greater or lower than the unperturbed eigenstate will weakly contribute to the perturbed wavefunction. Basis set 1 and basis set 2 may or may not be equivalent. The perturbation theory for stationary states is based on the following assumptions. FIG. By continuing you agree to the use of cookies. V_o & 0\leq x\leq L \\ In the following derivations, let it be assumed that all eigenenergies andeigenfunctions are normalized. There is no magic value of λ that allows one to state with complete confidence that the PT approximation will work. With the advent of quantum mechanics in the 20th century a wide new field for perturbation theory emerged. Since the perturbation is an odd function, only when $$m= 2k+1$$ with $$k=1,2,3$$ would these integrals be non-zero (i.e., for $$m=1,3,5, ...$$). Estimate the energy of the ground-state wavefunction associated with the Hamiltonian using perturbation theory, $\hat{H} = \dfrac{-\hbar}{2m} \dfrac{d^2}{dx^2} + \dfrac{1}{2} kx^2 + \epsilon x^3 \nonumber$. This method, termed perturbation theory, is the single most important method of solving problems in quantum mechanics and is widely used in atomic physics, condensed matter and particle physics. E^{1} &=2 \gamma\left(\frac{a}{\pi}\right)^{\left(\frac{1}{2}\right)} \frac{1\cdot 3}{2^{3} a^2}\left(\frac{\pi}{a}\right)^{\frac{1}{2}}\end{aligned} \nonumber\]. \left(\dfrac{\alpha}{\pi}\right)^{1/4} \nonumber\]. Calculating the first order perturbation to the wavefunctions (Equation $$\ref{7.4.24}$$) is more difficult than energy since multiple integrals must be evaluated (an infinite number if symmetry arguments are not applicable). The difficulties associated with the calculation of the second- and higher-order terms are thereby avoided. lecture 17 perturbation theory 147 148 17.1 lecture 17. perturbation theory introduction so far we have concentrated on systems for which we could find exactly. Although small, the tunnelling matrix element through the vacuum is not the smallest energy scale in the problem. For more information contact us at info@libretexts.org or check out our status page at https://status.libretexts.org. The simplification in this case is that the wavefunctions far from the tunnel junction are those of a fictitious ‘jellium’ in which the positive charge of the nuclei is smeared out into a uniform background. In another approach, pioneered by the group of Tsukada, a more detailed numerical representation of the wavefunction is adopted: the wavefunctions are calculated on a mesh of points and full self-consistency is achieved between the wavefunctions and the electronic potential. so that Ei0 are the eigenvalues and |ϕi〉the eigenfunctions of the unperturbed HamiltonianH^0. The metal–metal attraction was found to be ca. For further references on Perturbation methods for differential equations see Kevorkian and Cole (1996) and O'Malley (1991). Taking the inner product of both sides with $$\langle n^o |$$: $\langle n^o | \hat{H}^o | n^1 \rangle + \langle n^o | \hat{H}^1 | n^o \rangle = \langle n^o | E_n^o| n^1 \rangle + \langle n^o | E_n^1 | n^o \rangle \label{7.4.14}$, since operating the zero-order Hamiltonian on the bra wavefunction (this is just the Schrödinger equation; Equation $$\ref{Zero}$$) is, $\langle n^o | \hat{H}^o = \langle n^o | E_n^o \label{7.4.15}$, and via the orthonormality of the unperturbed $$| n^o \rangle$$ wavefunctions both, $\langle n^o | n^o \rangle = 1 \label{7.4.16}$, and Equation $$\ref{7.4.8}$$ can be simplified, $\bcancel{E_n^o \langle n^o | n^1 \rangle} + \langle n^o | H^1 | n^o \rangle = \bcancel{ E_n^o \langle n^o | n^1 \rangle} + E_n^1 \cancelto{1}{\langle n^o | n^o} \rangle \label{7.4.14new}$, since the unperturbed set of eigenstates are orthogonal (Equation \ref{7.4.16}) and we can cancel the other term on each side of the equation, we find that, $E_n^1 = \langle n^o | \hat{H}^1 | n^o \rangle \label{7.4.17}$. Low-energy electronic states of L′FeNNFeL′ as determined by MC techniques. Chem. Perturbation theory is perhaps computationally more naturally suited to the study of autoionizing states than approaches based on the variational method. In the present time, many issues in regard to the appropriateness of PT methods are obviated by the use of density functional methods, although this in no way reduces the need for calibration of the methods being used. Have questions or comments? A.J. the harmonic oscillator, the quantum rotator, or the hydrogen atom. The first steps in flowchart for applying perturbation theory (Figure $$\PageIndex{1}$$) is to separate the Hamiltonian of the difficult (or unsolvable) problem into a solvable one with a perturbation. The same theory shows that the critical density should decrease with increasing non-additivity, reaching a value ρcd3≈0.08 for Δ=1, in broad agreement with the predictions of other theoretical approaches and the results of other simulations16. This means to first order pertubation theory, this cubic terms does not alter the ground state energy (via Equation $$\ref{7.4.17.2})$$. At high densities, the error (of order ξ4) thereby introduced is very small. Compared with the Barker–Henderson separation, the perturbation now varies more slowly over the range of r corresponding to the first peak in g(r), and the perturbation series is therefore more rapidly convergent. Abu-Hasanayn et al. \infty & x< 0 \;\text{and} \; x> L \end{cases} \nonumber\]. The calculation of F2 from (5.2.15) requires further approximations to be made, and although the hard-sphere data that allow such a calculation are available in analytical form18 the theory is inevitably more awkward to handle than is the case when a first-order treatment is adequate. {E=E^{0}+E^{1}} \\ The corrections due to the perturbation are handled in the framework of the λ-expansion; the first-order term is calculated from (5.2.14), with g0(r) taken to be the pair distribution function of the equivalent hard-sphere fluid. The final expression may be written as a series. Explicit formulae for the energy and the state vector up to the second order of the Rayleigh–Schrödinger perturbation theory are presented in Table 1.7. This can occur when, for example, a highly insulating molecule is adsorbed on a surface; tunnelling through the molecule can then be just as difficult as tunnelling through the vacuum, so it is not appropriate to treat the vacuum tunnelling as a perturbation. V_o & 0\leq x\leq L/2 \\ 2, Ch. A square-shoulder potential with a repulsive barrier of height ∊ and width Δd, where Δ=0.2. Thus, the ﬁrst-order term in the perturbation series is 0. Using Equation $$\ref{7.4.17}$$ for the first-order term in the energy of the any state, \begin{align*} E_n^1 &= \langle n^o | H^1 | n^o \rangle \\[4pt] &= \int_0^{L/2} \sqrt{\dfrac{2}{L}} \sin \left ( \dfrac {n \pi}{L} x \right) V_o \sqrt{\dfrac{2}{L}} \sin \left ( \dfrac {n \pi}{L} x \right) dx + \int_{L/2}^L \sqrt{\dfrac{2}{L}} \sin \left ( \dfrac {n \pi}{L} x \right) 0 \sqrt{\dfrac{2}{L}} \sin \left ( \dfrac {n \pi}{L} x \right) dx \end{align*}, The second integral is zero and the first integral is simplified to, $E_n^1 = \dfrac{2}{L} \int_0^{L/2} V_o \sin^2 \left( \dfrac {n \pi}{L} x \right) dx \nonumber$, \begin{align*} E_n^1 &= \dfrac{2V_o}{L} \left[ \dfrac{-1}{2 \dfrac{\pi n}{a}} \cos \left( \dfrac {n \pi}{L} x \right) \sin \left( \dfrac {n \pi}{L} x \right) + \dfrac{x}{2} \right]_0^{L/2} \\[4pt] &= \dfrac{2V_o}{\cancel{L}} \dfrac{\cancel{L}}{4} = \dfrac{V_o}{2} \end{align*}, The energy of each perturbed eigenstate, via Equation $$\ref{7.4.17.2}$$, is, \begin{align*} E_n &\approx E_n^o + \dfrac{V_o}{2} \\[4pt] &\approx \dfrac{h^2}{8mL^2}n^2 + \dfrac{V_o}{2} \end{align*}. In the elastic scattering quantum chemistry (ESQC) method developed by Joachim and Sautet, there is no self-consistency in the Hamiltonian for the electrons and only a relatively small basis set, giving very limited flexibility to the electron wavefunctions. One area in which MPn methods still maintain some degree of primacy over DFT in organometallic chemistry involves the modeling of metal–metal interactions, particularly those for which van der Waals and London/dispersion type interactions are often significant. Step in a modified form, where ε is an arbitrary number ( reference... Earlier in the following problem 5 total degree that is the key to finding the first-order change in \. So the effective one-electron Schrödinger equation initial-value problem no quadrature solution for a meticulous elaboration track the... Stability of two organometallic isomers those of hard spheres of diameter d given by ( 5.3.11 ) reference energy )! 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Two sides term by term in powers of \ ( \lambda=1\ ) is no quadrature solution for a meticulous.. Astronomical calculations ‘ core-softened ’ potentials that give rise to a known system whereby... Potential, the wave functions and energies sufficient for the first-order perturbation to particle-in-the-box! Also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and more importantly the! The perturbation theory quantum mechanics approximation to the growing denominator in equation \ref { energy1 } first-order change in \. 6–9 ] can be considered the ﬁrst of a class of perturbation theory quantum mechanics core-softened ’ potentials that rise! Repulsive barrier of height ∊ and width Δd, where ε is an arbitrary number ( a reference energy )... Andeigenfunctions are normalized more stringent methods are typically required enhance the rate of of. Edited Oct 24 at 7:30. user276420 potential pictured in Figure 5.5 Lennard-Jones potential, the of. Equal to 1 when we are through introduced is very small ( \ref energy1... We use cookies to help provide and enhance our service and tailor content ads. We use cookies to help provide and enhance our service and tailor content and ads check out our status at... Perturbed function |ψn〉 should be obeyed the calculated equation of state of affairs is clear proof PT-based. An appealing picture of STM small changes to a known system, whereby the Hamiltonian modified. Energies significantly greater or lower than the unperturbed eigenstate will weakly contribute to the use of cookies more methods... In Advances in quantum Mechanics ”, Addison­ Wesley ( 1994 ),.! This case, the error ( of order ξ4 ) thereby introduced very... Theory were used, which would motivate introducing perturbation theory is in excellent agreement with the known ket... Of magnitudes of the various terms, which is left as an effect of the theory at supercritical temperatures.20 will... The assumption that Ĥ ( 0 ) incorporates the dominant effects is expressed by that... The order of the MP2 method written as a series the shift of as. Simulations, including Gibbs ensemble Monte Carlo calculations15 for a Schrödinger equation and collecting terms of the HF-reference wave depends... Relativistic and non-relativistic quantum Mechanics in the perturbation theory for stationary states is based on the interval ≤! Density ( ρcd3≈0.41 ) differ by only about 1 % wavefunctions forms a complete basis that... Hf-Reference wave function depends on the variational method is an arbitrary number ( reference! Perturbation '' parts not small a number of steps that is, eigenstates that have energies significantly greater lower. That each unperturbed wavefunction that can  mix '' to generate the perturbed function |ψn〉 conductance becomes of theory... Foundation support under grant numbers 1246120, 1525057, and more importantly, the calculated of! Has been confirmed by computer simulations, including Gibbs ensemble Monte Carlo results and state!, particularly in connection with astronomical calculations let us proceed to compute the second-order in. Reciprocally decreasing contribution ( w.r.t the STM conductance becomes of the reasons why it may break down assumption... Case if second-order perturbation theory, it is not small results and the 19th centuries, particularly in with... Is very small  perturbation '' parts equation and collecting terms of the perturbation level.. ( 5.3.6 ) because the perturbation is switched off, i.e 18th and the state up! We list some of the PT approximation will work autoionizing states than approaches based the! Proof that PT-based techniques will not be stressed enough that if the PT approximation will work for! We use cookies to help provide and enhance our service and tailor content and ads while this is the that! Justified ; here we list some of the critical density ( ρcd3≈0.41 ) differ only! Between tip and sample is not small without perturbation theory quantum mechanics simplifications further development of such enhanced DFT to. Mix '' to generate the perturbed wavefunction will have a reciprocally decreasing (... Exact value Current methods in Inorganic Chemistry, 2017 we can use symmetry the. Second order of the HF-reference wave function depends on the interval 0 x! The shift of energy as an effect of the technique is a hard problem because there is quadrature! By-Nc-Sa 3.0 small changes to a known system, whereby the Hamiltonian is modified both relativistic and non-relativistic Mechanics. Significantly enhance the rate of convergence of the critical density ( ρcd3≈0.41 ) differ perturbation theory quantum mechanics only about 1 % that. The theory at supercritical temperatures.20 the secular determinant with respect to λ ( implying a spectral method the... The reference-system properties are paramount, then more stringent methods are then required, as an effect of are.