A measured vibrational line gives an energy difference. Halving that line is an exact ground-state correction only for a specified harmonic oscillator. When adjacent vibrational spacings are unequal, a model must connect those differences to the energy above the potential minimum.
The short answer is to name the level model and the transition assignments before calculating. We will infer a two-term anharmonic correction from two assigned spacings, test what a third spacing can reveal, and show how higher-order terms and calibration affect the answer. Every spectrum and coefficient in the worked examples is fictional. The article explains an inference, not a measurement of a named molecule or a complete spectroscopic analysis.
eduKateSG · Zero-Point Energy
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Choose the question closest to your task, or follow the original worked model in order. All teaching chapters remain open.
ROUTE 1 · CHAPTERS 1–6
Define the spectroscopic inference
State the reference, level expression and transition assignments.
ROUTE 2 · CHAPTERS 7–12
Fit and challenge the model
Reconstruct levels and test an original cubic counterexample.
ROUTE 3 · CHAPTERS 13–18
Track calibration and uncertainty
Separate model validation, shared errors and unit conventions.
ROUTE 4 · CHAPTERS 19–26
Transfer and evaluate evidence
Check physical limits, changed spectra and inference ledgers.
ROUTE 5 · CHAPTERS 27–31
Test residuals and continue
Bound omitted terms and choose the next evidence route.
Chapter contents
Define the spectroscopic inference · Chapters 1–6
1. Separate a transition from an energy reference
2. State the two-term anharmonic model
3. Define which two spacings are supplied
Fit and challenge the model · Chapters 7–12
7. Work an original two-spacing record
8. Reconstruct the level ledger
9. Use the third spacing as a model check
10. Add a cubic term explicitly
11. Construct two models with the same supplied lines
12. Derive what a third spacing identifies in the cubic model
Track calibration and uncertainty · Chapters 13–18
13. Distinguish model identification from model validation
14. Keep calibration offsets visible
15. Separate a scale error from an offset
16. Propagate bounded spacing errors
Transfer and evaluate evidence · Chapters 19–26
19. Distinguish vibrational term values from rotational lines
20. Avoid turning the low-level polynomial into an infinite spectrum
21. Compare an isotope change under an explicit potential model
22. Use the primary spectroscopic sources for their scope
23. Try a changed-spectrum problem
24. Check the changed-spectrum answers
Test residuals and continue · Chapters 27–31
27. Test a third-spacing residual with uncertainty
28. Preserve dependence in a corrected three-spacing estimate
29. Bound a neglected cubic contribution conditionally
CHAPTER 1 OF 31 · Define the spectroscopic inference
1. Separate a transition from an energy reference
Let G(v) denote a vibrational term value expressed in reciprocal centimetres, with the potential minimum chosen as zero. Multiplying a term value by the appropriate hc conversion gives an energy. The vibrational quantum number v labels the levels relevant to the model.
A transition between two levels measures their difference, not either level’s absolute term value. Adding the same constant to all G values leaves every transition unchanged. Therefore a reference-relative ground correction requires a model linking the spectrum to the potential minimum.
Our teaching model specifies that relationship explicitly. It does not imply that two observed peaks determine an arbitrary energy baseline. When a reported zero-point correction is inferred from spectroscopy, keep the measured differences, the fitted level expression and the chosen potential reference as three related but distinct parts of the account. That separation is the starting point for understanding why a harmonic half-line estimate can require correction.
CHAPTER 2 OF 31 · Define the spectroscopic inference
2. State the two-term anharmonic model
Use G(v) = A(v + one-half) − B(v + one-half)² for the low vibrational levels under discussion. A and B are expressed in reciprocal centimetres. We take A positive and B nonnegative in the illustrative cases, so the spacings decrease with increasing v over the chosen range.
The harmonic special case is B = zero. Then adjacent spacings are all A and the ground term is A/two. Positive B changes both the spacing pattern and the relationship between the first spacing and G(0).
This polynomial expression is a stated low-level spectroscopic model. It must not be extended to every arbitrarily large integer as a globally stable spectrum: with positive B, it eventually turns downward. A real molecule has a finite bound vibrational ladder and a dissociation region, while higher terms and other interactions can matter. Our calculations concern the assigned low levels for which the stipulated expression is being tested.
CHAPTER 3 OF 31 · Define the spectroscopic inference
3. Define which two spacings are supplied
Call s0 the spacing G(1) − G(0) and s1 the spacing G(2) − G(1). The subscripts identify the lower vibrational level of each adjacent transition. Both values must refer to the same mode, molecular state and relevant isotopic species in a physical interpretation.
A first observed line and a second observed line are not automatically these two adjacent spacings. They could include rotational structure, another mode, another species, an overtone or a transition from a thermally populated level. The algebra below assumes that the assignments have already been justified.
In a paper exercise, the assignments are supplied so the inference can be checked cleanly. In an evidence review, record how they were established and which corrections define the vibrational spacing being used. A precise peak position does not compensate for a wrong level label. The quantity entering the equation must match its definition before the arithmetic can answer the intended question.
At v = zero, G(0) = A/two − B/four. At v = one, G(1) = three A/two − nine B/four. Subtracting gives s0 = A − two B.
The fundamental spacing therefore differs from the harmonic coefficient A when B is nonzero. Calling an observed fundamental a harmonic frequency without qualification would mix a measured level difference with a model coefficient.
The distinction is visible in the subtraction: the quadratic term changes by eight B/four, or two B, between these levels. Do the level calculation before relying on a memorised relation. It supplies a route for checking signs and factors and makes clear which convention for the anharmonic coefficient is being used. Another source can label its coefficients differently, so the displayed level expression should remain the common reference when comparing formulas.
CHAPTER 5 OF 31 · Define the spectroscopic inference
5. Derive the second spacing and solve the coefficients
At v = two, G(2) = five A/two − twenty-five B/four. Subtracting G(1) gives s1 = A − four B. Hence s0 − s1 = two B.
The inferred coefficients are B = (s0 − s1)/two and A = two s0 − s1. These are exact algebraic consequences of the two-term model and the supplied assignments. Two independent spacings identify two coefficients within that model.
They do not validate the truncation of the level expansion. Any pair of values can be substituted into these expressions, but the resulting coefficients must still respect the intended domain and reproduce additional evidence if the model is to be used beyond the two fitted differences. Fitting two coefficients to two exact readings leaves no unused spacing with which to test the assumed form. A third assigned difference can provide a useful consistency check.
CHAPTER 6 OF 31 · Define the spectroscopic inference
6. Infer the ground correction
Substitute A and B into G(0) = A/two − B/four. The result is G0,inferred = (seven s0 − three s1)/eight. This is a reference-relative ground correction under the stated two-term expression.
The harmonic half-fundamental estimate is s0/two. Their difference is three(s0 − s1)/eight, or three B/four. When B is positive, the two-term inferred ground correction is greater than half the first spacing.
This does not mean that anharmonicity always raises a ground energy relative to every possible harmonic reference. The harmonic coefficient A/two is another comparison, and G(0) = A/two − B/four lies below it. The direction depends on what is held fixed: the fitted harmonic coefficient or the measured first spacing. State the comparator rather than saying only that an anharmonic correction is positive or negative.
CHAPTER 7 OF 31 · Fit and challenge the model
7. Work an original two-spacing record
Take s0 = 1,200 and s1 = 1,120 reciprocal centimetres. Their difference is eighty, so B = forty. Then A = 2,400 − 1,120 = 1,280 reciprocal centimetres.
The ground correction is 640 − ten = 630 reciprocal centimetres. Equivalently, (seven times 1,200 minus three times 1,120)/eight gives 630. Halving the first spacing would give 600, thirty below this model’s inferred value.
Every number belongs to the fictional level model. The result is not a newly measured molecular constant or an exact correction for every potential with similar-looking lines. Its value is that another reader can reconstruct the coefficients, level values and original spacings. Keeping that forward check attached to the inverse calculation makes clear what has been established and which physical assumptions still require evidence.
For A = 1,280 and B = forty, the first four model levels are as follows. All term values and spacings are in reciprocal centimetres relative to the stipulated potential minimum.
| Vibrational level v | v + one-half | Model term G(v) | Spacing from preceding level |
|---|---|---|---|
| zero | 0.5 | 630 | Not applicable |
| one | 1.5 | 1,830 | 1,200 |
| two | 2.5 | 2,950 | 1,120 |
| three | 3.5 | 3,990 | 1,040 |
The first two differences reproduce the supplied record. The third is a prediction of the fitted two-term model, not another input used to choose its coefficients. This distinction is useful when planning a check: an observation already used in the fit confirms arithmetic consistency, while an additional assigned spacing can test a consequence that was not forced by the initial two equations.
CHAPTER 9 OF 31 · Fit and challenge the model
9. Use the third spacing as a model check
In the two-term expression, the next spacing is s2 = G(3) − G(2) = A − six B. Consequently s2 = two s1 − s0. For the original record, that is 2,240 − 1,200 = 1,040 reciprocal centimetres.
The adjacent spacings form an arithmetic sequence within this model. Their second finite difference s2 − two s1 + s0 is zero. A nonzero value beyond justified uncertainty signals that the exact two-term account does not reproduce all three assigned differences.
That discrepancy does not identify its cause uniquely. Higher-order vibrational terms, assignment errors, unresolved rotational effects, species mixtures or calibration issues could contribute. The model check supplies a specific inconsistency to investigate. It should not be converted directly into a claim about one microscopic mechanism without the additional evidence needed to distinguish those explanations.
Consider the extended expression G(v) = Az − Bz² + Cz³, where z = v + one-half. The first two spacings now become s0 = A − two B + thirteen C/four and s1 = A − four B + forty-nine C/four.
The ground term is A/two − B/four + C/eight. There are now three unknown coefficients but only two supplied spacings. The earlier two-equation inversion no longer identifies all of them.
This is a model change, not a small numerical correction that can be inserted without revisiting identifiability. A two-term fit can still reproduce the same two differences exactly while giving a different inferred ground correction from an allowed cubic model. The next chapter constructs that ambiguity with numbers, so the limitation is visible through substitution rather than only a general warning that higher-order effects might matter.
CHAPTER 11 OF 31 · Fit and challenge the model
11. Construct two models with the same supplied lines
The original two-term model has A = 1,280, B = forty and C = zero, giving G(0) = 630. Now choose a different low-level model with C = eight, A = 1,326 and B = seventy-six.
Its first spacing is 1,326 − 152 + twenty-six = 1,200. Its second is 1,326 − 304 + ninety-eight = 1,120. Thus it reproduces exactly the same two supplied line differences.
Its ground term is 663 − nineteen + one = 645 reciprocal centimetres. The two models agree on both fitted observations but disagree by fifteen on the reference-relative ground correction. This is an original counterexample to claiming that the two spacings determine G(0) without a level-form assumption. Better precision on those same two numbers would not remove this particular model ambiguity. An additional independent relationship is needed.
CHAPTER 12 OF 31 · Fit and challenge the model
12. Derive what a third spacing identifies in the cubic model
The third adjacent spacing is s2 = A − six B + 109 C/four. Forming s2 − two s1 + s0 cancels A and B and leaves six C. Therefore C = (s2 − two s1 + s0)/six.
For the extended example, s2 is 1,326 − 456 + 218 = 1,088. Its second difference is 1,088 − 2,240 + 1,200 = forty-eight, giving C = eight as stipulated.
With C known, the first two equations determine A and B within the three-term model. The corresponding ground expression can be written G(0) = (nineteen s0 − sixteen s1 + five s2)/sixteen. Substituting the extended record gives 645. A third spacing has supplied the extra relationship needed by the enlarged coefficient set, but it still does not prove that every still-higher term is absent.
CHAPTER 13 OF 31 · Track calibration and uncertainty
13. Distinguish model identification from model validation
Two spacings determine two coefficients in the truncated model. Three determine three coefficients in the cubic model. Those counts explain why the displayed inversions are possible, but they do not certify the physical completeness of either expression.
An additional observation not already used to fit the chosen coefficients can test a prediction. Independent knowledge about the potential, state assignment or size of neglected terms can also support an interpretation. The evidence should address the assumption that matters, rather than only increase the number of digits in a fitted coefficient.
A useful report states the model order, the data used to infer its coefficients and any observations held aside for checking. It distinguishes algebraic exactness within a model from agreement with an actual spectrum. That practice makes the ground correction reproducible while keeping its physical confidence proportionate to the available evidence and its modelling limits.
CHAPTER 14 OF 31 · Track calibration and uncertainty
14. Keep calibration offsets visible
Suppose both measured spacings contain the same additive offset b, so sobserved = strue + b for each. The inferred B from their difference is unchanged. The inferred A increases by b, and the two-term ground correction increases by b/two.
The shared offset cancels from the anharmonic spacing difference but does not cancel from the full ground-correction combination. In the original example, a common offset of four reciprocal centimetres changes the inferred correction by two, even though the inferred B remains forty.
This is a concrete reason to distinguish a difference-based coefficient from a weighted combination of absolute spacing values. A cancellation valid for one derived quantity need not hold for another. The example supplies a calibration model deliberately; it does not establish that a real instrument has a common offset or that such an offset is the only source of error. The appropriate calibration relationship must be supported for the actual observations.
CHAPTER 15 OF 31 · Track calibration and uncertainty
15. Separate a scale error from an offset
For an affine calibration sobserved = a strue + b with positive scale a, the inferred two-term coefficients transform as Bobserved = a Btrue and Aobserved = a Atrue + b. The inferred ground correction is therefore a Gtrue + b/two under the same level model.
A scale error multiplies every spacing and its inferred energy scale. An offset adds equally to the measured spacings but affects their weighted combination according to its coefficient sum. These effects should not be conflated when comparing two records.
If a and b are independently known, the observed spacings can be transformed to the intended calibration basis before fitting. If they are unknown, more peaks do not automatically identify them along with every physical coefficient. The calibration and spectral model can have parameter trade-offs. Record which quantities are established independently and which are estimated from the same data, rather than counting one fitted correction twice as if it were additional evidence.
CHAPTER 16 OF 31 · Track calibration and uncertainty
16. Propagate bounded spacing errors
The two-term correction is linear in s0 and s1. If their absolute errors are bounded by δ0 and δ1, a conservative deterministic bound on the correction error is (seven δ0 + three δ1)/eight.
For δ0 = two and δ1 = three reciprocal centimetres, this is twenty-three/eight, or 2.875. The worst direction has the two errors pushing the weighted combination in the same direction: one positive and the other negative because their coefficients have opposite signs.
This is an interval bound, not a standard deviation or a confidence level. If the errors are linked, the allowed joint combinations may be narrower than the independent rectangle used here. A common offset, for example, produces a smaller coefficient sum than the sum of absolute coefficients. Preserve the relationship between errors when it is known. Otherwise a mathematically careful propagation can still misrepresent the information supplied by the calibration record.
CHAPTER 17 OF 31 · Track calibration and uncertainty
17. Propagate variances with their covariance
If the spacing estimates have variances σ0² and σ1² and covariance Cov(s0,s1), the variance of the inferred correction is [forty-nine σ0² + nine σ1² − forty-two Cov(s0,s1)]/sixty-four.
For σ0 = two, σ1 = three and correlation one-half, the covariance is three. The numerator is 196 + eighty-one − 126 = 151, giving a standard deviation √151/eight, approximately 1.536. If the two estimates were independent instead, the numerator would be 277 and the standard deviation approximately 2.080.
The smaller value in the correlated example follows from the supplied covariance and opposite weights. It is not a universal claim that correlation improves precision. A different sign or error structure changes the result. Also, neither variance calculation includes an unquantified error from omitting a cubic or higher term. Statistical uncertainty within a chosen model and uncertainty about that model are different questions.
CHAPTER 18 OF 31 · Track calibration and uncertainty
18. Keep units and wavenumber conventions explicit
The symbols A, B, C and G in this article use reciprocal centimetres. They are term values on an energy-divided-by-hc scale. A spacing in hertz, angular frequency or reciprocal metres must be converted consistently before combining it with these coefficients.
Multiplying a reciprocal-centimetre value by hc requires the corresponding conversion from centimetres to metres when SI constants are used. The numerical value in reciprocal metres is one hundred times the value in reciprocal centimetres. That unit conversion does not change the physical energy.
Ratios and coefficient identities can look dimensionally harmless while hiding a mixed-unit input. A useful ledger retains the original observed unit, the converted spacing and the formula convention used by the source. Do not substitute an angular frequency into a formula for an ordinary frequency or a spectroscopic wavenumber merely because each describes vibration. Their relationships contain specific factors, and those factors belong in the inference.
CHAPTER 19 OF 31 · Transfer and evaluate evidence
19. Distinguish vibrational term values from rotational lines
Actual molecular spectra can include rotational structure associated with a vibrational transition. A single resolved line can differ from the pure vibrational band origin because it connects specified rotational states as well as vibrational states.
Our s0 and s1 are assigned vibrational spacings after whatever observation model and corrections justify those quantities. They are not arbitrary neighbouring lines on a plotted spectrum. The NIST molecular formulation separates vibrational and rotational contributions, which is one reason the full assignment matters.
A learner does not need to perform a complete rotational analysis to understand this boundary. The practical task is to label what the supplied number represents. If a question gives a vibrational spacing, use it as such. If it gives a raw line frequency, identify the missing relationship before treating it as G(v + one) − G(v). Correct algebra cannot repair an observation that was assigned to the wrong physical quantity.
CHAPTER 20 OF 31 · Transfer and evaluate evidence
20. Avoid turning the low-level polynomial into an infinite spectrum
With positive B and no higher term, A(v + one-half) − B(v + one-half)² eventually decreases as v grows. Extending that expression over all nonnegative integers would not describe a globally bounded oscillator spectrum.
The model is therefore used over a stated low-level range or as part of a more complete potential description. An exact Morse potential has additional structure connecting its finite bound ladder and dissociation behaviour, but a two-term fit to two spacings does not by itself establish that the molecular potential is exactly Morse.
This distinction matters if someone uses the fitted coefficients to infer a dissociation energy or count every bound level. Those conclusions require the extra potential assumptions and their domain conditions. Our ground-correction exercise does not make them automatically. A useful low-level fit remains useful when its range is stated honestly; it becomes misleading when its convenient polynomial is promoted to a complete physical potential without supporting evidence.
CHAPTER 21 OF 31 · Transfer and evaluate evidence
21. Compare an isotope change under an explicit potential model
As a separate teaching assumption, suppose a fixed Morse-type potential gives harmonic coefficient A proportional to the inverse square root of reduced mass and anharmonic coefficient B proportional to its inverse. This is a conditional scaling model, not an exact rule for every molecule or isotopic environment.
Starting from A = 1,280 and B = forty, multiply the reduced mass by four. The scaled coefficients are A = 640 and B = ten. The new two-term ground correction is 320 − 2.5 = 317.5 reciprocal centimetres.
That is not exactly half the original 630, which would be 315, because the two coefficients scale differently. The example extends the root’s isotope discussion by showing where a simple frequency-ratio argument stops. A real comparison also needs the appropriate potential, isotopic species, mode identity and treatment of other changes. State the scaling assumptions before using an isotope shift to infer a correction.
CHAPTER 22 OF 31 · Transfer and evaluate evidence
22. Use the primary spectroscopic sources for their scope
NIST’s diatomic spectral database states the vibrational expansion in powers of v + one-half and distinguishes its coefficients from rotational parameters. NIST’s vibrational notes also explain why measured fundamentals differ from harmonic frequencies and why sufficient level information matters.
The original records here use invented coefficients and assigned spacings. Their two- and three-term inversions, calibration examples and uncertainty ledgers are not values reported for a named molecule. They follow from the displayed level expressions and supplied error models.
Read NIST's molecular energy-level definitions and its notes on vibrational frequencies and scaling, checked on 25 October 2026. For a primary account of deriving molecular zero-point values from spectroscopic constants, see Irikura's diatomic zero-point-energy study. Keep its real molecular analysis distinct from the deliberately small fictional model used for this lesson.
CHAPTER 23 OF 31 · Transfer and evaluate evidence
23. Try a changed-spectrum problem
A new fictional two-term record supplies s0 = 1,500 and s1 = 1,440 reciprocal centimetres. Infer A, B and G(0). Compare the result with half the first spacing and predict s2 under the same truncated expression.
Then suppose both supplied spacings contain a known common positive offset of six. Calculate the correction inferred from the offset readings and explain which fitted coefficient changes. Finally consider independent bounded spacing errors of one and two reciprocal centimetres and obtain a conservative bound for the ground-correction error.
Work through the level differences before reading the next chapter. This problem changes the numerical record and the calibration conditions while retaining the same algebraic structure. The transfer is complete only when the answer identifies the reference, transition assignments and model order, rather than presenting one number labelled zero-point energy with no account of how the spectrum supported it.
CHAPTER 24 OF 31 · Transfer and evaluate evidence
24. Check the changed-spectrum answers
The spacing difference is sixty, so B = thirty. Then A = 3,000 − 1,440 = 1,560. The ground correction is 780 − 7.5 = 772.5 reciprocal centimetres. Half the fundamental is 750, which is 22.5 below the two-term result, matching three B/four.
The predicted third spacing is two times 1,440 minus 1,500 = 1,380. Adding a common offset of six changes the inferred A to 1,566 while leaving B = thirty. The inferred ground correction rises by three to 775.5.
For independent error ranges bounded by one and two, the conservative absolute correction error is (seven + six)/eight = 1.625 reciprocal centimetres. This bound concerns the supplied spacing errors within the two-term model. It does not cover an unknown higher-order contribution. A successful solution preserves that limitation and reconstructs the original spacings from the inferred coefficients as a final forward check.
CHAPTER 25 OF 31 · Transfer and evaluate evidence
25. Build an observation-to-correction ledger
Begin with species and state identity, mode assignment, transition labels, observed units and calibration basis. State which quantities are raw lines and which have been interpreted as vibrational spacings. Then display the chosen level expression and its range of use.
Record the fitted coefficients and the observations used to determine them. Keep a third-spacing prediction separate from the two fitted inputs. If an uncertainty is reported, identify whether it comes from bounded errors, a variance model, shared calibration or an estimate of neglected physical terms.
Finally reconstruct the supplied differences from the inferred levels. This checks the inverse arithmetic without pretending that a fit validates itself. The ledger helps a reader locate the earliest missing relationship: an assignment, calibration, model-order assumption or reference connection. That is more informative than treating every discrepancy as a vague correction factor or adding more digits to a model that has not been tested.
CHAPTER 26 OF 31 · Transfer and evaluate evidence
26. Repair, stabilise and extend the method
If a fundamental is automatically halved, return to G(0), G(1) and their difference and ask whether B was assumed zero. If two measured lines are used as adjacent spacings, verify their assignments. If a two-term fit is called exact for the molecule, construct the cubic counterexample and identify the additional evidence needed.
To stabilise the calculation, derive A and B, reconstruct the level ledger and predict the third spacing. Check the harmonic limit s0 = s1, which returns B = zero and G(0) = s0/two. Then change the numerical record and the calibration model independently.
For extension, add a justified higher-order term and count the new unknowns before fitting. Propagate the actual joint uncertainty and preserve the potential-reference convention. The transferable habit is to keep measured differences and inferred baseline corrections connected by an explicit model, rather than letting a familiar half-frequency formula conceal the assumptions that made it applicable.
CHAPTER 27 OF 31 · Test residuals and continue
27. Test a third-spacing residual with uncertainty
Let D = s2 − two s1 + s0. The exact two-term model predicts D = zero. If each spacing has an independently supplied absolute error bound of two reciprocal centimetres, the residual error is bounded by two + four + two = eight.
For the original first two spacings, a third reading of 1,045 gives D = five. Its interval includes zero, so that record does not exclude the two-term prediction under the stated error bounds. A third reading of 1,088 gives D = forty-eight, whose interval from forty to fifty-six excludes zero.
For D = forty-eight, the residual correction is five D/sixteen = fifteen, taking the two-term estimate from 630 to 645.
Within the cubic extension, the latter residual implies C between forty/six and fifty-six/six, approximately 6.667 to 9.333. This is conditional on the assignments, error ranges and model order. Rejecting a two-term account does not prove that a cubic term is the only missing physical effect. The residual is a specific testable mismatch, and its interpretation must remain connected to the assumptions that made the test meaningful.
CHAPTER 28 OF 31 · Test residuals and continue
28. Preserve dependence in a corrected three-spacing estimate
In the cubic expression, G(0) = (nineteen s0 − sixteen s1 + five s2)/sixteen. If all three spacing errors have absolute bounds of two, a conservative correction-error bound is (thirty-eight + thirty-two + ten)/sixteen = five reciprocal centimetres.
The extended fictional record therefore gives 645 plus or minus five under those input ranges and that model. Computing the two-term estimate and the residual correction separately can be useful algebraically, but their errors are linked because both reuse s0 and s1. Treating them as independent random observations would introduce an unjustified uncertainty model.
The same principle applies to covariance propagation. Work directly with the complete linear coefficient vector and the joint covariance matrix when the inputs are shared. A rearranged formula does not create new independent evidence. This is why an uncertainty ledger should preserve the original spacing observations and their relationships, even when the final correction is presented through a convenient intermediate coefficient.
CHAPTER 29 OF 31 · Test residuals and continue
29. Bound a neglected cubic contribution conditionally
If a two-term fit is applied to spacings generated by the cubic model, its inferred ground term differs from the cubic ground term by minus fifteen C/eight. This follows by substituting the cubic spacings into the two-term formula, not by assuming the correction is small.
An independently justified bound |C| no greater than four would therefore bound this particular truncation effect by 7.5 reciprocal centimetres. Such a statement addresses one specified omitted term. It does not cover arbitrary higher-order functions, a wrong line assignment or an unknown calibration error.
Without a justified bound on C or additional spectral information, the two spacings alone do not supply a numerical truncation-error guarantee. Selecting a small value because the two-term answer looks plausible would be circular. The useful next evidence concerns the missing curvature or potential relationship, rather than repeating the same two fitted spacings with more printed digits. A conditional error bound is valuable when its source is explicit and remains attached to the model it actually constrains.
Is half a fundamental always a zero-point correction?
It is exact for the declared harmonic oscillator. Under the two-term anharmonic model, G(0) is greater than half the first spacing by three B/four when B is positive. Other models require their own relationship.
Do two accurate spacings determine the exact ground correction?
They determine A and B within the stated two-term expression. They do not exclude higher terms or resolve an arbitrary energy reference. The cubic counterexample fits the same two spacings with a different inferred correction.
Does a third spacing prove the model is complete?
No. It can test a two-term prediction or determine a third coefficient in an enlarged expression. Further neglected terms, assignments and observation effects still need evidence appropriate to the physical question.
Can a small statistical error bar include model error automatically?
No. A variance propagated through a fixed formula concerns the specified input-error model. An omitted cubic term or wrong assignment is a different source of uncertainty and requires its own investigation or bound.
CHAPTER 31 OF 31 · Test residuals and continue
31. Continue through the relevant spectroscopic routes
Return to Zero-Point Energy for reference states, harmonic spectra and the role of molecular corrections. Use Molecular Physics and Spectroscopy for the connection between molecular motion, transitions and observations.
For the difference between a calculated harmonic frequency and a measured fundamental, continue with Computational Chemistry. For a separate way to constrain a ground energy from a specified nonharmonic Hamiltonian, read Gaussian variational bounds for a quartic oscillator. Its trial-state inequality is a different inference from fitting assigned spacings.
For the related issue of a preparation’s level populations, read why mean oscillator energy does not determine the ground-state fraction. For how interactions change harmonic normal modes, compare coupled-mode ground energies under different Hamiltonian controls. Transition assignment and a fitted vibrational correction should remain distinct from both a population claim and a coupling reconstruction.
Continue through Learning That Transfers into Life and How X Works. A clear explanation states the assignments, level expansion and reference before presenting an inferred ground correction.