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Complete expressions of the thermal-expansion coefficient $\alpha$ and theGr\"{u}neisen parameter $\Gamma$ are derived on the basis of theself-consistent renormalization (SCR) theory. By considering zero-point as wellas thermal spin fluctuation under the stationary condition, the specific heatfor each class of the magnetic quantum critical point (QCP) specified by thedynamical exponent $z=3$ (FM) and $z=2$ (AFM) and the spatial dimension ($d=3$and $2$) is shown to be expressed as $C_{V}=C_a-C_b$, where $C_a$ is dominantat low temperatures, reproducing the past SCR criticality endorsed by therenormalization group theory. Starting from the explicit form of the entropyand using the Maxwell relation, $\alpha=\alpha_a+\alpha_b$ (with $\alpha_a$ and$\alpha_b$ being related to $C_a$ and $C_b$, respectively) is derived, which isproven to be equivalent to $\alpha$ derived from the free energy. Thetemperature-dependent coefficient found to exist in $\alpha_b$, which isdominant at low temperatures, contributes to the crossover from thequantum-critical regime to the Curie-Weiss regime and even affects the quantumcriticality at 2d AFM QCP. Based on these correctly calculated $C_{V}$ and$\alpha$, Gr\"{u}neisen parameter $\Gamma=\Gamma_a+\Gamma_b$ is derived, where$\Gamma_a$ and $\Gamma_b$ contain $\alpha_a$ and $\alpha_b$, respectively. Theinverse susceptibility coupled to the volume $V$ in $\Gamma_b$ gives rise todivergence of $\Gamma$ at the QCP for each class even though characteristicenergy scale of spin fluctuation $T_0$ is finite at the QCP, which gives afinite contribution in $\Gamma_a=-\frac{V}{T_0}\left(\frac{\partialT_0}{\partial V}\right)_{T=0}$. General properties of $\alpha$ and $\Gamma$including their signs as well as the relation to $T_0$ and the Kondotemperature in temperature-pressure phase diagrams of Ce- and Yb-based heavyelectron systems are discussed.

Gr\"{u}neisen Parameter and Thermal Expansion near Magnetic Quantum Critical Points in Itinerant Electron Systems